Factorization of Algebraic Expressions

Why A is correct:
Expand (x + 2y)(4x + 3): you get 4x² + 3x + 8xy + 6y, which rearranges to 4x² + 8xy + 3x + 6y ✓

Why others are wrong:
- B: (2x + y)(2x + 3) = 4x² + 6x + 2xy + 3y — doesn't match (wrong coefficients for xy and y terms)
- C: (4x + 3y)(x + 2) = 4x² + 8x + 3xy + 6y — has 8x instead of 3x, and 3xy instead of 8xy
- D: (x + y)(4x + 6) = 4x² + 6x + 4xy + 6y — missing the 8xy term and has wrong x coefficient

Why A is correct:
The greatest common factor (GCF) of all three terms is 2ab. When you factor it out: 2ab(a + 2b + 3). Expanding this gives 2a²b + 4ab² + 6ab ✓

Why the others are wrong:
- B: Uses ab instead of 2ab as the GCF—you'd miss the factor of 2
- C: If you expand 2ab(a + b + 3), you get 2a²b + 2ab² + 6ab, which doesn't match (middle term is wrong)
- D: This isn't fully factored; it's still partially expanded and leaves 2a²b outside the brackets

Why A is correct:
Group the terms as (3xy + 6x) + (2y + 4). Factor out 3x from the first group and 2 from the second: 3x(y + 2) + 2(y + 2). Now (y + 2) is a common factor, giving (y + 2)(3x + 2). Expand to check: (y + 2)(3x + 2) = 3xy + 2y + 6x + 4 ✓

Why the others are wrong:
- B: (x + 2)(3y + 2) = 3xy + 2x + 6y + 4 — wrong terms (has 2x and 6y, not 6x and 2y)
- C: (3x + y)(y + 2) = 3xy + 6x + y² + 2y — has y² which isn't in the original
- D: (3x + 2y)(y + 2) = 3xy + 6x + 2y² + 4y — has y² and 4y which aren't in the original

A. (a + b)(x + y) ✓

Expand it: (a + b)(x + y) = ax + ay + bx + by. Perfect match!

Why the others are wrong:
- B. (a + x)(b + y) expands to ab + ay + bx + xy—doesn't match
- C. (a + y)(b + x) expands to ab + ax + by + xy—doesn't match
- D. (a + b)(x - y) expands to ax - ay + bx - by—the signs are wrong

Quick tip: Group the first two terms (ax + ay = a(x + y)) and the last two (bx + by = b(x + y)), then factor out the common (x + y).

Why A is correct:
Group the terms: (2m² + 4m) + (6mn + 12n). Factor each group: 2m(m + 2) + 6n(m + 2). Now (m + 2) is a common factor, giving (m + 2)(2m + 6n). Factor out 2 from the second bracket: 2(m + 2)(m + 3n). ✓

Why others are wrong:
- B: Expands to 2m² + 6mn + 4mn + 12n = 2m² + 10mn + 12n (wrong middle term)
- C: Expands to 2m² + 2mn + 4m + 4n (wrong coefficients for mn and n terms)
- D: Same as B, just written in different order—still doesn't match the original expression

Why A is correct:
Group the terms as (5x² + 10xy) + (x + 2y). Factor out 5x from the first group to get 5x(x + 2y), and factor out 1 from the second to get 1(x + 2y). Now (x + 2y) is a common factor, leaving (x + 2y)(5x + 1). Expand to check: 5x² + x + 10xy + 2y ✓

Why the others are wrong:
- B: (x + y)(5x + 2) expands to 5x² + 2x + 5xy + 2y — the coefficients don't match our original.
- C: (x + 2)(5x + y) expands to 5x² + xy + 10x + 2y — wrong terms entirely.
- D: (x + 2y)(x + 5) expands to x² + 5x + 2xy + 10y — the x² coefficient should be 5, not 1.

# Explanation

Why A is correct:
Group the terms as (6ab + 4b) + (9a² + 6a). Factor each group: 2b(3a + 2) + 3a(3a + 2). Now (3a + 2) is a common factor, giving (2b + 3a)(3a + 2). Check: expanding gives 6ab + 4b + 9a² + 6a ✓

Why B is wrong:
Expanding (3a + 2)(2a + 3b) gives 6a² + 9ab + 4a + 6b—the 9a² term is wrong (should be 6a² instead).

Why C is wrong:
The minus sign is incorrect. Expanding (2b + 3a)(3a - 2) gives 6ab - 4b + 9a² - 6a, which doesn't match the original.

Why D is wrong:
Expanding (3a + b)(2a + 2) gives 6a² + 6a + 2ab + 2b—missing the 9a² and 4b terms entirely.

Correct answer: (x + 2)(x - 5)

When you expand this: (x + 2)(x - 5) = x² - 5x + 2x - 10 = x² - 3x - 10 ✓

Why others are wrong:
- (x - 9)² expands to x² - 18x + 81 (wrong middle and constant term)
- (x + 9)(x - 1) expands to x² + 8x - 9 (wrong middle term and constant)
- x(x - 9) expands to x² - 9x (missing the constant term -10)

Why A is correct:
Expand (2x + 1)(x + 3): 2x·x + 2x·3 + 1·x + 1·3 = 2x² + 6x + x + 3 = 2x² + 7x + 3 ✓

Why others are wrong:
- B: (2x + 3)(x + 1) = 2x² + 2x + 3x + 3 = 2x² + 5x + 3 (middle term is 5x, not 7x)
- C: (x + 2)(2x + 2) = 2x² + 2x + 4x + 4 = 2x² + 6x + 4 (wrong middle term and constant)
- D: (x + 3)(x + 2) = x² + 5x + 6 (this is a different quadratic entirely—no 2x² term)

Why A is correct:
Group the terms in pairs: (3x² - 6xy) + (-2x + 4y). Factor each pair: 3x(x - 2y) - 2(x - 2y). Both groups share the common factor (x - 2y), so you get (x - 2y)(3x - 2).

Why others are wrong:
- B: (x - 2y)(x + 3) expands to x² + 3x - 2xy - 6y, which doesn't match the original.
- C: (3x - 2y)(x - 2) expands to 3x² - 6x - 2xy + 4y—the coefficient of x is wrong (should be -2, not -6).
- D: (x + 2y)(3x + 2) has the wrong signs and expands to 3x² + 2x + 6xy + 4y.

Why A is correct:
Group the terms: (6a² + 9ab) + (-4a - 6b). Factor each group: 3a(2a + 3b) - 2(2a + 3b). Now (2a + 3b) is common, so you get (2a + 3b)(3a - 2). Check by expanding: 6a² - 4a + 9ab - 6b ✓

Why the others are wrong:
- B: Expands to 6a² + 4a + 9ab + 6b (wrong signs on the constant terms)
- C: Same as A, just written in different order—both are correct, but A is the listed answer
- D: Expands to 6a² + 4ab - 9ab - 6b (wrong middle term and signs don't match)

Why A is correct:
Group the terms in pairs: (4xy - 6x) + (2y - 3). Factor out 2x from the first pair to get 2x(2y - 3), and factor out 1 from the second to get 1(2y - 3). Now (2y - 3) is a common factor, so you get (2y - 3)(2x + 1). Check: expand it back and you get 4xy + 2y - 6x - 3 ✓

Why others are wrong:
- B: The sign is wrong—expanding (2y + 3)(2x + 1) gives 4xy + 2y + 6x + 3, which doesn't match.
- C: Expanding (4x + 2)(y - 3) gives 4xy - 12x + 2y - 6, which is different.
- D: This isn't fully factored; you could factor out more from (4xy + 2y).

Why A is correct:
When you expand (x + 2)(x + 3), you get x² + 3x + 2x + 6 = x² + 5x + 6. The two numbers (2 and 3) multiply to give 6 and add to give 5, which matches the original expression.

Why the others are wrong:
- B: (x - 2)(x - 3) = x² - 5x + 6 — the middle term is negative, not positive
- C: (x + 1)(x + 6) = x² + 7x + 6 — the numbers add to 7, not 5
- D: (x - 1)(x - 6) = x² - 7x + 6 — the middle term is negative and wrong size

Why A is correct:
Expand (3x + 1)(x - 1): you get 3x² - 3x + x - 1 = 3x² - 2x - 1 ✓

Why others are wrong:
- B: (3x - 1)(x + 1) = 3x² + 3x - x - 1 = 3x² + 2x - 1 (wrong middle term—it's positive, not negative)
- C: (x + 1)(x - 3) = x² - 2x - 3 (wrong leading coefficient—should be 3, not 1)
- D: (3x + 1)(x + 1) = 3x² + 3x + x + 1 = 3x² + 4x + 1 (completely wrong)

Why A is correct:
Group the terms: (6xy - 9x) + (-4y + 6). Factor each group: 3x(2y - 3) - 2(2y - 3). Now (2y - 3) is a common factor, giving (2y - 3)(3x - 2). Check: (2y - 3)(3x - 2) = 6xy - 4y - 9x + 6 ✓

Why the others are wrong:
- B: (2y - 3)(3x + 2) = 6xy + 4y - 9x - 6 — wrong signs on the last two terms
- C: (2y + 3)(3x - 2) = 6xy - 4y + 9x - 6 — wrong signs throughout
- D: (3x + 2)(2y - 1) = 6xy - 3x + 4y - 2 — completely different result

Why A is correct:
Group the terms: (2a² - 3ab) + (-2a + 3b). Factor out *a* from the first group and -1 from the second: a(2a - 3b) - 1(2a - 3b). Now (2a - 3b) is a common factor, giving (2a - 3b)(a - 1). Check: expand to get 2a² - 2a - 3ab + 3b ✓

Why the others are wrong:
- B: Expanding (a - 3b)(2a + 1) gives 2a² + a - 6ab - 3b, which doesn't match.
- C: Expanding (2a + 3b)(a - 1) gives 2a² - 2a + 3ab - 3b; the middle term has the wrong sign.
- D: Expanding (2a - 3b)(a + 1) gives 2a² + 2a - 3ab - 3b; the coefficient of *a* is wrong.

Why A is correct:
Multiply it out: (4x + 3)(2x - 5) = 8x² - 20x + 6x - 15 = 8x² - 14x - 15 ✓

Why the others are wrong:
- B: (4x - 3)(2x + 5) = 8x² + 20x - 6x - 15 = 8x² + 14x - 15 (wrong sign on the middle term)
- C: (2x - 3)(4x + 5) = 8x² + 10x - 12x - 15 = 8x² - 2x - 15 (wrong middle coefficient)
- D: Same as C—just reordered factors, so it gives 8x² - 2x - 15 (wrong middle coefficient)

Why A is correct:
Expand (x - 4)(x - 1)(x + 1): First, (x - 1)(x + 1) = x² - 1. Then (x - 4)(x² - 1) = x³ - x - 4x² + 4 = x³ - 4x² - x + 4 ✓

Why the others are wrong:
- B: (x + 4)(x - 1)(x + 1) = (x + 4)(x² - 1) = x³ + 4x² - x - 4. The x² term is +4, not -4.
- C: (x + 2)(x² - 2) only gives two factors, not three. Expanding gives x³ + 2x² - 2x - 4, which doesn't match.
- D: (x + 4)(x - 4) = x² - 16, a quadratic—the original is cubic (degree 3), so this can't be right.

Why A is correct:
Group the terms as (3xy - 6y) + (-x + 2). Factor out 3y from the first group to get 3y(x - 2), and factor out -1 from the second group to get -1(x - 2). Now (x - 2) is a common factor, so you get (x - 2)(3y - 1). Check: (x - 2)(3y - 1) = 3xy - x - 6y + 2 ✓

Why the others are wrong:
- B: (x - 2)(3y + 1) = 3xy + x - 6y - 2 — wrong signs
- C: (x + 2)(3y - 1) = 3xy - x + 6y - 2 — wrong signs on both groups
- D: (x - 2)(3y + 2) = 3xy + 2x - 6y - 4 — wrong coefficient and constant term

Why A is correct:
Group the first three terms: a² - 2ab + b² = (a - b)². Now you have (a - b)² - c², which is a difference of squares. This factors as [(a - b) - c][(a - b) + c].

Why the others are wrong:
- B: Expands to a² - ac - ab + bc - ab + b² + bc - c², which doesn't match the original.
- C: This would give (a - b)²c + (a - b)², missing the c² term entirely.
- D: This isn't factorised—it's still an expression to simplify, not a product of factors.

Why A is correct:
- Factor the numerator: $x^2 - y^2 = (x+y)(x-y)$ (difference of squares)
- Factor the denominator: $x^2 - 2xy + y^2 = (x-y)^2$ (perfect square trinomial)
- Simplify: $\frac{(x+y)(x-y)}{(x-y)^2} = \frac{x+y}{x-y}$ (cancel one $(x-y)$)

Why others are wrong:
- B: Reversed the factors; you'd get this if you flipped numerator and denominator
- C: Incorrect; the $(x+y)$ term doesn't cancel completely
- D: This is just rewriting the original problem, not simplifying it

Correct Answer: A ($2yz$)

To divide fractions, flip the second fraction and multiply:
$$\frac{9y^2}{12z} \times \frac{8z^2}{3y}$$

Multiply across: $\frac{9y^2 \cdot 8z^2}{12z \cdot 3y} = \frac{72y^2z^2}{36yz}$

Simplify by canceling: $\frac{72}{36} = 2$, $y^2 ÷ y = y$, $z^2 ÷ z = z$ → $2yz$

Why others are wrong:
- B ($yz$): Missing the coefficient 2; forgot to simplify $72 ÷ 36$
- C ($2y$): Forgot the $z$ term (didn't simplify the $z$ variables correctly)
- D ($6yz$): Used $72 ÷ 12 = 6$ instead of $72 ÷ 36 = 2$

Why A is correct:
Factor the numerator: $x^2 - 5x + 6 = (x - 2)(x - 3)$
Factor the denominator: $x^2 - x - 6 = (x - 3)(x + 2)$
Cancel the common factor $(x - 3)$: $\frac{(x-2)(x-3)}{(x-3)(x+2)} = \frac{x-2}{x+2}$

Why the others are wrong:
- B: Incorrectly factored the numerator or forgot to cancel $(x-3)$
- C: Swapped numerator and denominator factors
- D: Only true if all factors cancelled, but $(x-2)$ and $(x+2)$ don't cancel

# Explanation

Why A is correct:
Find a common denominator for all three fractions. The LCD of 4a, 6a, and 3a is 12a.
- $\frac{3}{4a} = \frac{9}{12a}$
- $\frac{5}{6a} = \frac{10}{12a}$
- $\frac{1}{3a} = \frac{4}{12a}$

Now combine: $\frac{9}{12a} + \frac{10}{12a} - \frac{4}{12a} = \frac{15}{12a} = \frac{5}{4a}$ (simplified by dividing by 3)

Why B is wrong:
This is the unsimplified version. Since 15 and 12 share a common factor of 3, you must reduce it to $\frac{5}{4a}$.

Why C is wrong:
This would require the numerators to add up to 6 (so $\frac{6}{12a}$ reduces to $\frac{1}{2a}$), but $9 + 10 - 4 = 15$, not 6.

Why D is wrong:
This assumes you add all three numerators instead of subtracting the last one: $9 + 10 + 4 = 23$, but the correct operation is $9 + 10 - 4 = 15$.

# Explanation

Why A is correct:
To change the denominator from $4b$ to $12b$, multiply both numerator and denominator by 3 (since $4b × 3 = 12b$). This gives $\frac{3a × 3}{4b × 3} = \frac{9a}{12b}$.

Why the others are wrong:
- B multiplies only the denominator by 3, forgetting to multiply the numerator—this changes the value of the fraction.
- C multiplies the numerator by 2 and denominator by 3, which is inconsistent and doesn't preserve the fraction's value.
- D multiplies the numerator by 4 and denominator by 3, again inconsistent and wrong.

# Explanation

Why A is correct:
All fractions have the same denominator variable (x), so combine the numerators: 2/3x + 5/6x - 1/2x. Find a common denominator of 6x: (4/6x) + (5/6x) - (3/6x) = (4 + 5 - 3)/6x = 6/6x = 1/x. ✓

Why the others are wrong:
- B (2/x): You'd get this if you forgot to subtract 1/2x or made an arithmetic error with the numerators.
- C (1/2x): This is only part of the answer—you'd get this if you only added the first two fractions and ignored the third.
- D (6/6x): This is an intermediate step before simplifying; 6/6x reduces to 1/x, not the final answer.

# Explanation

Why A is correct:
Multiply the fractions: $\frac{3x \times 8y^2}{4y \times 9x^2} = \frac{24xy^2}{36x^2y}$

Cancel common factors: $x$ cancels once, $y$ cancels once, and $\frac{24}{36} = \frac{2}{3}$

Result: $\frac{2y}{3x}$ ✓

Why the others are wrong:
- B ($\frac{y}{2x}$): Wrong denominator coefficient—you'd get this if you incorrectly simplified the 24/36 part.
- C ($\frac{2y}{3x^2}$): Fails to cancel one $x$ from numerator and denominator; you only cancel what appears in both.
- D ($\frac{8x}{12y}$): This doesn't simplify the original fractions at all and leaves variables in wrong positions.

# Explanation

Why A is correct:
- Factor the numerator: $2x^2 - 18x = 2x(x - 9)$
- Factor the denominator: $4x - 36 = 4(x - 9)$
- Cancel the common factor $(x - 9)$: $\frac{2x(x-9)}{4(x-9)} = \frac{2x}{4} = \frac{x}{2}$

Why the others are wrong:
- B: Forgetting to factor out the 2 from the numerator; this leaves an uncancelled $(x-9)$
- C: This reverses the fraction or incorrectly cancels terms
- D: This comes from only canceling $2x$ from the numerator with 4 from the denominator, ignoring the $(x-9)$ factors

Correct answer: A — (y²)/(x)

To divide fractions, flip the second fraction and multiply:
$$\frac{5x}{2y} \times \frac{6y^3}{15x^2}$$

Multiply across: $\frac{5x \cdot 6y^3}{2y \cdot 15x^2} = \frac{30xy^3}{30x^2y}$

Cancel common factors (30, x, and y): $\frac{y^2}{x}$ ✓

Why others are wrong:
- B — Flips the exponents; comes from incorrectly inverting the final answer
- C — Doesn't cancel enough; forgets to reduce $x^2$ and leaves an extra $y$
- D — Only cancels one power of $y$; misses that $y^3 ÷ y = y^2$

# Explanation

Why A is correct:
All fractions share denominator *b*, so focus on the numerators with denominator 15*b*:
- Convert to common denominator 15*b*: $\frac{5a}{15b} + \frac{6a}{15b} - \frac{a}{15b}$
- Combine: $\frac{5a + 6a - a}{15b} = \frac{10a}{15b}$
- Simplify by dividing both numerator and denominator by 5: $\frac{2a}{3b}$ ✓

Why B is wrong:
This is the unsimplified intermediate step. You must reduce $\frac{10a}{15b}$ further by canceling the common factor of 5.

Why C is wrong:
This would be correct if the second term were subtracted instead of added, giving $\frac{5a - 6a - a}{15b} = \frac{-2a}{15b}$, which doesn't match anyway.

Why D is wrong:
This results from incorrectly combining terms—it's too large. The correct answer has *b* in the denominator, not as part of the coefficient.

Why A is correct:
Group the terms: $(8x + 4y) + (2xz + yz)$. Factor each group: $4(2x + y) + z(2x + y)$. Now $(2x + y)$ is a common factor, giving $(2x + y)(4 + z)$. Check: $(2x + y)(4 + z) = 8x + 2xz + 4y + yz$ ✓

Why others are wrong:
- B: $(x + 2y)(4 + z) = 4x + xz + 8y + 2yz$ — doesn't match the original
- C: $(2x + y)(2 + z) = 4x + 2xz + 2y + yz$ — the coefficients of $x$ and $y$ terms are wrong
- D: $(4x + 2y)(1 + z) = 4x + 4xz + 2y + 2yz$ — coefficients don't match

Correct Answer: A

Group the terms in pairs: $(5a + 15b) + (2ac + 6bc)$. Factor each group: $5(a + 3b) + 2c(a + 3b)$. Now $(a + 3b)$ is a common factor, so you get $(a + 3b)(5 + 2c)$. Check by expanding: $(a + 3b)(5 + 2c) = 5a + 2ac + 15b + 6bc$ ✓

Why others are wrong:

• B: $(5a + 15b)(2 + c) = 10a + 5ac + 30b + 15bc$ — the coefficients don't match the original.
• C: $(a + 3b)(2 + 5c) = 2a + 5ac + 6b + 15bc$ — wrong coefficients for the first and third terms.
• D: $(a + 2c)(5 + 3b) = 5a + 3ab + 10c + 6bc$ — introduces $ab$ terms that aren't in the original.

Why A is correct:
Group the terms as $(x^3 + 2x^2y) + (xz + 2yz)$. Factor out $x^2$ from the first group and $z$ from the second: $x^2(x + 2y) + z(x + 2y)$. Now $(x + 2y)$ is a common factor, giving $(x + 2y)(x^2 + z)$. ✓

Why the others are wrong:
- B: Expanding $(x + y)(x^2 + 2z) = x^3 + 2xz + x^2y + 2yz$ — this doesn't match the original expression (wrong coefficient on $x^2y$).
- C: $(x + 2y)(x + z) = x^2 + xz + 2xy + 2yz$ — this is only degree 2, not degree 3 like the original.
- D: $(x^2 + y)(x + 2z) = x^3 + 2x^2z + xy + 2yz$ — has $2x^2z$ instead of $2x^2y$, so it doesn't match.

Why A is correct:
The greatest common factor of all three terms is $5q$. When you factor it out: $15p^2q^2 ÷ 5q = 3p^2q$, $-10pq ÷ 5q = -2p$, and $5q ÷ 5q = 1$. This gives $5q(3p^2q - 2p + 1)$.

Why the others are wrong:
- B: Only factors out $5$, leaving $q$ inside the bracket—this doesn't fully factor the expression.
- C: Factors out $5p$ instead of $5q$, which doesn't divide evenly into the last term ($5q$).
- D: Has the wrong signs ($+2p$ and $-1$) inside the bracket; if you expand it, you won't get the original expression.

A is correct: Find the greatest common factor of all terms: $8u^2v$, $12uv^2$, and $4uv$ share $4uv$. Factoring it out gives $4uv(2u + 3v + 1)$. You can verify by expanding: $4uv \cdot 2u + 4uv \cdot 3v + 4uv \cdot 1 = 8u^2v + 12uv^2 + 4uv$ ✓

B is wrong: Only factors out $4u$, not the full GCF; the terms inside don't match the original when expanded.

C is wrong: Only factors out $4v$; again, incomplete GCF and the bracketed terms don't simplify to the original.

D is wrong: Has the correct GCF ($4uv$) but uses a minus sign instead of plus—this expands to $8u^2v - 12uv^2 + 4uv$, which is incorrect.

Correct Answer (A):
- The GCF of all three terms is 5xy (largest factor that divides each term)
- Dividing each term by 5xy: $20x^3y^2 ÷ 5xy = 4x^2y$, $-15x^2y ÷ 5xy = -3x$, $5xy ÷ 5xy = 1$
- Result: $5xy(4x^2y - 3x + 1)$ ✓

Why others are wrong:
- B: Uses 5x as GCF, but y is also a common factor, so this isn't fully factored
- C: Uses 5y as GCF, but x is also a common factor; the bracket doesn't match the original expression
- D: Has the wrong signs inside the bracket (should be minus, not plus); expanding this would give different terms

Why A is correct:
- First, find the GCF of all three terms: 6kl is the largest factor common to 18k²l, 12kl², and 6kl
- Divide each term by 6kl: (18k²l ÷ 6kl) + (12kl² ÷ 6kl) - (6kl ÷ 6kl) = 3k + 2l - 1
- So the factorisation is 6kl(3k + 2l - 1)

Why the others are wrong:
- B: Only factors out 6k, not 6kl (leaves l inside the bracket unnecessarily)
- C: Only factors out 6l, not 6kl (leaves k inside the bracket unnecessarily)
- D: The signs are wrong inside the brackets—should be + 2l and - 1, not - 2l and + 1

Why A is correct:
First, find the greatest common factor (GCF) of all terms: 16t³, 24t², and 8t share a GCF of 8t. Factoring out 8t gives 8t(2t² − 3t + 1). You can verify by expanding: 8t × 2t² = 16t³, 8t × (−3t) = −24t², and 8t × 1 = 8t. ✓

Why the others are wrong:
- B: Wrong signs in the bracket—expanding 8t(2t² + 3t − 1) gives 16t³ + 24t² − 8t, which doesn't match.
- C: Only factors out 8, not the full GCF of 8t; this isn't fully factored.
- D: Factors out only t, leaving 8 still inside the bracket; not fully factored since 8 is a common factor of the remaining terms.

Why A is correct:
The greatest common factor of all three terms is $7rs$. When you factor this out: $21r^2s ÷ 7rs = 3r$, $-14rs^2 ÷ 7rs = -2s$, and $7rs ÷ 7rs = 1$. This gives $7rs(3r - 2s + 1)$. You can verify by expanding: $7rs \times 3r = 21r^2s$ ✓, $7rs \times (-2s) = -14rs^2$ ✓, $7rs \times 1 = 7rs$ ✓

Why others are wrong:
- B: Only factors out $7r$, leaving $s$ inside the brackets—not fully factored
- C: Only factors out $7s$, leaving $r$ inside the brackets—not fully factored
- D: Has wrong signs inside the brackets ($+2s$ instead of $-2s$); expanding would give $-14rs^2$ as $+14rs^2$

Why A is correct:
The greatest common factor of all three terms is $3ab$. Factoring it out: $27a^3b ÷ 3ab = 9a^2$, $-9a^2b^2 ÷ 3ab = -3ab$, and $3ab^3 ÷ 3ab = b^2$. This gives $3ab(9a^2 - 3ab + b^2)$. ✓

Why the others are wrong:
- B: Has a $+$ instead of $-$ for the middle term; doesn't match the original expression.
- C: Only factors out $3a$, leaving $b$ unfactored in the first term ($9a^2b$ instead of $9a^2$).
- D: Only factors out $3b$, leaving $a$ unfactored in the first term ($9a^3$ instead of $9a^2$).

Why A is correct:
Expand $(x + 5y)(x + 2z)$: you get $x^2 + 2xz + 5xy + 10yz$, which matches the original expression exactly.

Why the others are wrong:
- B: $(x + 2y)(x + 5z) = x^2 + 5xz + 2xy + 10yz$ — the $xy$ and $xz$ coefficients are swapped (gives 2xy and 5xz instead of 5xy and 2xz)
- C: $(x + 5y)(2x + z) = 2x^2 + xz + 10xy + 5yz$ — creates a $2x^2$ term, which doesn't exist in the original
- D: Same as B, just reordered — still has the wrong coefficients on $xy$ and $xz$

Why A is correct:
Group the terms as $(4m + mn) + (4n + n^2)$. Factor out $m$ from the first group: $m(4 + n)$. Factor out $n$ from the second group: $n(4 + n)$. Now $(4 + n)$ is a common factor, giving $(m + n)(4 + n)$.

Why the others are wrong:
- B: Expands to $4m + m^2 + 4n + mn$, which doesn't match the original.
- C: Expands to $4m^2 + 4mn + mn + n^2 = 4m^2 + 5mn + n^2$, which is wrong.
- D: Expands to $mn + 4m + 4n^2 + 16n$, which doesn't match the original.

Why A is correct:
Expand $(2p + 3q)(3p + r)$: you get $6p^2 + 2pr + 9pq + 3qr$, which matches the original expression exactly.

Why the others are wrong:
- B: $(3p + 2q)(2p + r) = 6p^2 + 3pr + 4pq + 2qr$ — the coefficients of $pr$ and $qr$ don't match.
- C: $(6p + 3q)(p + r) = 6p^2 + 6pr + 3pq + 3qr$ — again, wrong coefficients for $pr$ and $pq$.
- D: $(2p + 3r)(3p + q) = 6p^2 + 2pq + 9pr + 3qr$ — the $pq$ and $pr$ terms are swapped compared to the original.

Correct answer (A):
Group the terms: $(5m^2n + 10mn) + (3m + 6)$. Factor each group: $5mn(m + 2) + 3(m + 2)$. Now $(m + 2)$ is the common factor, giving $(m + 2)(5mn + 3)$. Check by expanding: $(m + 2)(5mn + 3) = 5m^2n + 3m + 10mn + 6$ ✓

Why others are wrong:
- B: Expanding gives $5m^2 + 3mn + 10m + 6$—the first term should be $5m^2n$, not $5m^2$.
- C: Expanding gives $5m^2n - 10mn + 3m - 6$—wrong signs; this has minus instead of plus.
- D: Expanding gives $5m^2n + 10mn + 3m + 6$... wait, this also works! But check: the second factor should be $5mn + 3$, not $5mn + 2$, since we factored out 3 from $3m + 6$.

# Explanation

Why A is correct:
Expand $(3x + y)(3x + 4z)$: you get $9x^2 + 12xz + 3xy + 4yz$, which matches the original expression exactly.

Why B is wrong:
Expanding $(3x + y)(x + 4z)$ gives $3x^2 + 12xz + xy + 4yz$. The $x^2$ coefficient is 3, not 9, so this doesn't match.

Why C is wrong:
This isn't even fully factored—it's just grouping terms without finding common factors between the groups.

Why D is wrong:
Expanding $(3x - y)(3x + 4z)$ gives $9x^2 + 12xz - 3xy - 4yz$. The middle terms have negative signs, which doesn't match the original.

A is correct. Group the terms in pairs: $(2p^2 - 4pq) + (3pr - 6qr)$. Factor each group: $2p(p - 2q) + 3r(p - 2q)$. Now $(p - 2q)$ is a common factor, giving $(p - 2q)(2p + 3r)$. ✓

B is wrong: Has $(p + 2q)$ instead of $(p - 2q)$—if you expand it, you get $2p^2 + 4pq + 3pr + 6qr$, which has wrong signs.

C is wrong: $(p - 2q)(p + 2r)$ expands to $p^2 + 2pr - 2pq - 4qr$—the coefficient of $p^2$ is wrong (should be 2, not 1).

D is wrong: $(2p + 3r)(2p - 3q)$ expands to $4p^2 - 6pq + 6pr - 9qr$—multiple coefficients don't match the original.

A is correct:
Group the terms in pairs: $(7mn + 14n) + (3m + 6)$. Factor each group: $7n(m + 2) + 3(m + 2)$. Now $(m + 2)$ is a common factor, so you get $(m + 2)(7n + 3)$. ✓

B is wrong:
If you expand $(m + 2)(7n - 3)$, you get $7mn - 3m + 14n - 6$. The signs don't match the original expression (you'd have $-3m$ and $-6$, not $+3m$ and $+6$).

C is wrong:
$(7n + 3)(m - 2)$ expands to $7mn - 14n + 3m - 6$. The second and fourth terms have the wrong signs.

D is wrong:
$(7m + 3)(n + 2)$ expands to $7mn + 14m + 3n + 6$. This doesn't match—you get $14m$ and $3n$ instead of $14n$ and $3m$.

Why A is correct:
Use factoring by grouping: pair the terms as $(8ab - 4a) + (2b - 1)$. Factor out $4a$ from the first group to get $4a(2b - 1)$, then factor out $1$ from the second group to get $1(2b - 1)$. Now $(2b - 1)$ is a common factor, giving $(2b - 1)(4a + 1)$. You can verify by expanding: $(2b - 1)(4a + 1) = 8ab + 2b - 4a - 1$ ✓

Why the others are wrong:
- B: $(2b - 1)(a + 4) = 2ab + 8b - a - 4$ — doesn't match the original
- C: $(4a + 1)(2b + 1) = 8ab + 4a + 2b + 1$ — wrong signs (has $+4a$ and $+1$ instead of $-4a$ and $-1$)
- D: $(2b + 1)(4a - 1) = 8ab - 2b + 4a - 1$ — wrong signs (has $-2b$ and $+4a$ instead of $+2b$ and $-4a$)

Why A is correct:
Group the terms as $(x^3 + x^2y) + (-4x - 4y)$. Factor out $x^2$ from the first group and $-4$ from the second: $x^2(x + y) - 4(x + y)$. Now factor out the common $(x + y)$ to get $(x + y)(x^2 - 4)$. You can verify by expanding: $(x + y)(x^2 - 4) = x^3 - 4x + x^2y - 4y$ ✓

Why the others are wrong:
- B: Missing the $-4$ factor. Expanding gives $(x + y)(x - 2) = x^2 - 2x + xy - 2y$, which doesn't match.
- C: Has $x^2 + 4$ instead of $x^2 - 4$. Expanding gives $(x + y)(x^2 + 4) = x^3 + 4x + x^2y + 4y$ — wrong signs.
- D: Wrong grouping. Expanding gives $(x^2 - 4)(x - y) = x^3 - x^2y - 4x + 4y$ — the middle term has wrong sign.

Why C is correct:
Expand $(2a - 3)(3ab + 2)$: you get $2a(3ab) + 2a(2) - 3(3ab) - 3(2) = 6a^2b + 4a - 9ab - 6$, which matches the original expression exactly.

Why the others are wrong:
- A: Same factors as C but written in different order—actually equivalent to C, so also correct mathematically (likely a duplicate option)
- B: Expanding gives $6a^2b + 4a + 9ab + 6$—the signs are wrong (you get $+9ab$ instead of $-9ab$)
- D: Expanding gives $6a^2 + 4ab - 9a - 6b$—completely different; the $a^2$ term appears instead of $a^2b$

A is correct because $4x$ is the greatest common factor (GCF) of all three terms, and dividing each term by $4x$ gives: $8x^3 ÷ 4x = 2x^2$, $12x^2 ÷ 4x = 3x$, and $4x ÷ 4x = 1$.

B is wrong because the GCF is $4x$, not $2x$—you're left with a common factor that hasn't been fully removed.

C is wrong because the bracket doesn't match: multiplying $4x(2x^2 + x + 3)$ gives $8x^3 + 4x^2 + 12x$, not the original expression.

D is wrong because $4x^2$ is too large a factor (the last term $4x$ isn't divisible by $4x^2$), and the bracket simplifies incorrectly to $2x + 4$, which doesn't work.

Why A is correct:
The greatest common factor of all three terms is $5yz$. Factoring this out: $15y^2z ÷ 5yz = 3y$, $-10yz^2 ÷ 5yz = -2z$, and $5yz ÷ 5yz = 1$. So we get $5yz(3y - 2z + 1)$.

Why the others are wrong:
- B: Only factors out $5y$, leaving $z$ inside the bracket—incomplete factorization.
- C: Has a plus sign before $2z$, but the second term in the original is negative.
- D: Doesn't factor out the full common factor (missing the $y$), and the signs don't match when expanded.

A is correct: Find the greatest common factor (GCF) of all terms: 12m³, 18m², and 6m share a GCF of 6m. Factoring it out gives 6m(2m² - 3m + 1). You can verify by expanding: 6m × 2m² = 12m³, 6m × (−3m) = −18m², and 6m × 1 = 6m. ✓

B is wrong: The signs are incorrect inside the bracket—it shows +3m and −1, but dividing each term by 6m gives −3m and +1.

C is wrong: Doesn't factor out enough. The GCF is 6m, not just 6; you can still divide all terms inside by m.

D is wrong: Only factors out m, not the complete GCF. The terms inside (12m², −18m, 6) still share a common factor of 6 that should be pulled out.

Why A is correct:
The greatest common factor of all three terms is 3xy. Dividing each term by 3xy gives: $9x^2y ÷ 3xy = 3x$, $6xy^2 ÷ 3xy = 2y$, and $3xy ÷ 3xy = 1$. So $3xy(3x + 2y + 1)$ is correct.

Why the others are wrong:
- B: Only factors out $3x$, leaving $y$ inside the brackets—this doesn't fully extract the common factor.
- C: Only factors out $3y$, missing the $x$ that's common to all terms.
- D: Factors out $xy$ instead of $3xy$; the bracket still has a common factor of 3 that should be removed.

A is correct: First, find the greatest common factor (GCF) of all terms: GCF of 8u³, 12u², and 4u is 4u. Factoring this out gives 4u(2u² + 3u + 1). ✓

B is wrong: The middle term should be +3u, not –3u. Check: 4u(2u² – 3u + 1) = 8u³ – 12u² + 4u, which doesn't match the original.

C is wrong: While 2u is a common factor, it's not the *greatest* common factor—we can factor out 4u instead. Also, the bracket doesn't simplify fully.

D is wrong: The GCF is 4u, not 4u². Also, the bracket (2u + 3 + 1) doesn't make sense and doesn't equal the remaining expression.

Why A is correct:
Factor out the greatest common factor: all terms share $3b^3$. When you divide each term by $3b^3$, you get $9a^2 - 3a + 1$, giving $3b^3(9a^2 - 3a + 1)$. ✓

Why the others are wrong:
- B: Has a $+3a$ instead of $-3a$—wrong signs when factoring.
- C: Factors out $3a$ instead of $3b^3$—$3a$ doesn't divide the last term $3b^3$ evenly, and the bracket is messy/incorrect.
- D: Factors out $9b^3$, which is too large (the GCF is only $3b^3$)—this leaves you with $3a^2 - a + \frac{1}{3}$, not whole numbers.

Why A is correct:
Group the terms: $(x^2 + 2xy) + (3xz + 6yz)$. Factor each group: $x(x + 2y) + 3z(x + 2y)$. Now $(x + 2y)$ is a common factor, giving $(x + 2y)(x + 3z)$. Check by expanding: $(x + 2y)(x + 3z) = x^2 + 3xz + 2xy + 6yz$ ✓

Why the others are wrong:
- B: $(x + y)(x + 3z) = x^2 + 3xz + xy + 3yz$ — the $xy$ term is wrong (should be $2xy$)
- C: $(x + 2y)(3x + z) = 3x^2 + xz + 6xy + 2yz$ — has $3x^2$ instead of $x^2$
- D: $(x + 3y)(x + 2z) = x^2 + 2xz + 3xy + 6yz$ — has $2xz$ instead of $3xz$

A is correct.

The greatest common factor of all three terms is 7b (7 divides 14, 21, and 7; and *b* is in every term). Factoring this out: $14a^2b ÷ 7b = 2a^2$, $-21ab ÷ 7b = -3a$, and $7b ÷ 7b = 1$, giving $7b(2a^2 - 3a + 1)$.

Why others are wrong:
- B: Uses $7a$ as the common factor, but *a* isn't in the last term ($7b$).
- C: Has a minus sign instead of plus in the constant term; if you expand it, you get $7b(2a^2 - 3a - 1) = 14a^2b - 21ab - 7b$ ✗
- D: Has a plus sign instead of minus for the middle term; expanding gives $14a^2b + 21ab + 7b$ ✗

Why A is correct:
- The greatest common factor of all three terms (15r²s, 10rs, 5r) is 5r
- Dividing each term by 5r gives: 3rs − 2s + 1
- Checking: 5r(3rs − 2s + 1) = 15r²s − 10rs + 5r ✓

Why the others are wrong:
- B: The GCF is 5r, not 5s. If you factor out 5s, you can't get the first term 15r²s
- C: Uses decimals instead of whole numbers—messier and not standard form. Also, 10r is not the full GCF
- D: Has wrong signs (+ instead of −); expanding gives 15r²s + 10rs − 5r, which doesn't match the original

Why A is correct:
Expand $(x + 3y)(x + 2z)$: you get $x^2 + 2xz + 3xy + 6yz$, which matches the original expression perfectly.

Why the others are wrong:
- B: $(x + 2y)(x + 3z) = x^2 + 3xz + 2xy + 6yz$ — has $3xz$ and $2xy$ instead of $2xz$ and $3xy$
- C: Same as B (just reordered factors) — gives the wrong middle terms
- D: $(x + y)(x + z) = x^2 + xz + xy + yz$ — missing the coefficient 3 on the middle terms and doesn't match

Why A is correct:
Group the terms as $(6ab + 4b) + (9a + 6)$. Factor out $2b$ from the first group and $3$ from the second: $2b(3a + 2) + 3(3a + 2)$. Now $(3a + 2)$ is a common factor, giving $(2b + 3)(3a + 2)$. Expand to check: $2b(3a) + 2b(2) + 3(3a) + 3(2) = 6ab + 4b + 9a + 6$ ✓

Why the others are wrong:
- B & C: These are equivalent to A (just factors in different order), but the original expression has $6ab + 4b$ first, so factoring out $2b$ gives $(2b + 3)$ not $(3b + 2)$.
- D: Doesn't factor completely—$(6a + 6)$ can be reduced to $6(a+1)$, and this product doesn't equal the original expression when expanded.

A is correct. The greatest common factor of all three terms is $3xy$. Factoring this out gives: $3xy(9x^3y^2 ÷ 3xy + 6x^2y ÷ 3xy + 3xy ÷ 3xy) = 3xy(3x^2y + 2x + 1)$. Multiply back to verify: $3xy · 3x^2y = 9x^3y^2$ ✓, $3xy · 2x = 6x^2y$ ✓, $3xy · 1 = 3xy$ ✓

B is wrong: Uses $3x$ as the common factor, leaving $y$ terms inside the brackets that don't match the original expression.

C is wrong: Uses $3y$ as the common factor, but then $3y$ doesn't divide evenly into $6x^2y$; also leaves an $x$ term that shouldn't be there.

D is wrong: Has a minus sign ($-2x$) when the original term requires a plus sign ($+2x$).

Correct Answer: (a - b)(a + c)

Group the terms in pairs: $(a^2 - ab) + (ac - bc)$

Factor each pair: $a(a - b) + c(a - b)$

Factor out the common bracket: $(a - b)(a + c)$ ✓

Why others are wrong:

• B & C: Both expand to $a^2 - ac + ab - bc$, which has a $+ab$ instead of $-ab$.
• D: Expands to $a^2 - ac - ab + bc$, which has the wrong signs on the middle terms.

Why A is correct:
Group the terms as $(9x + 6) + (-3xy - 2y)$. Factor out 3 from the first group to get $3(3x + 2)$, and factor out $-y$ from the second group to get $-y(3x + 2)$. Now $(3x + 2)$ is a common factor: $(3x + 2)(3 - y)$. Check: expanding gives $9x + 6 - 3xy - 2y$ ✓

Why the others are wrong:
- B: Expanding $(3x - 2)(3 + y)$ gives $9x + 3xy - 6 - 2y$, which doesn't match.
- C: Expanding $(3x + 2)(y - 3)$ gives $3xy - 9x + 2y - 6$, which is the opposite order of terms and wrong signs.
- D: This doesn't properly factor—it's just grouping without finding the common binomial factor.

A is correct. Group the terms as $(4p^2 + 6pq) + (2rp + 3rq)$, factor out $2p$ from the first group and $r$ from the second to get $2p(2p + 3q) + r(2p + 3q)$, then factor out the common binomial $(2p + 3q)$ to get $(2p + 3q)(2p + r)$.

B is wrong: Expanding $(2p + 3r)(2p + q)$ gives $4p^2 + 2pq + 6pr + 3rq$—the middle terms don't match.

C is wrong: This doesn't factor correctly; expanding gives $4p^2 + 4pr + 6pq + 6qr$, which is different.

D is wrong: Expanding $(2p + r)(p + 3q)$ gives $2p^2 + 6pq + rp + 3rq$—the first term is $2p^2$, not $4p^2$.

Why A is correct:
Group the terms: $(7x^3 + 14x^2y) + (xz + 2yz)$. Factor out $7x^2$ from the first group and $z$ from the second: $7x^2(x + 2y) + z(x + 2y)$. Now $(x + 2y)$ is a common factor, giving $(x + 2y)(7x^2 + z)$. ✓

Why the others are wrong:
- B: Expanding $(7x + z)(x^2 + 2y)$ gives $7x^3 + 14xy + x^2z + 2yz$—the $x^2z$ term doesn't match the original.
- C: Expanding $(x + z)(7x^2 + 2y)$ gives $7x^3 + 2xy + 7x^2z + 2yz$—again, extra terms that aren't in the original.
- D: Expanding $(x + 2y)(x^2 + 7z)$ gives $x^3 + 7xz + 2x^2y + 14yz$—the coefficient of $x^3$ is 1, not 7.

Why A is correct:
Group the terms: $(10u + 5v) + (4uw + 2vw)$. Factor each group: $5(2u + v) + 2w(2u + v)$. Now $(2u + v)$ is a common factor, giving $(2u + v)(5 + 2w)$. Check by expanding: $10u + 4uw + 5v + 2vw$ ✓

Why the others are wrong:
- B: Expanding $(u + 2v)(5 + 4w) = 5u + 4uw + 10v + 8vw$ — doesn't match the original.
- C: Expanding $(2u + v)(5 + w) = 10u + 2uw + 5v + vw$ — the $w$ coefficients are wrong.
- D: Expanding $(5u + v)(2 + 2w) = 10u + 10uw + 2v + 2vw$ — doesn't match the original.

Why A is correct:
Group the terms in pairs: $(3a^2 + 12a) + (ab + 4b)$. Factor each group: $3a(a + 4) + b(a + 4)$. Now $(a + 4)$ is a common factor, giving $(a + 4)(3a + b)$. ✓

Why the others are wrong:
- B: $(3a + 4)(a + b) = 3a^2 + 3ab + 4a + 4b$ — the middle terms don't match (you get $3ab + 4a$, not $12a + ab$)
- C: $(a + 3)(4a + b) = 4a^2 + ab + 12a + 3b$ — wrong leading coefficient and constant term
- D: Same as A, just written in different order (commutative), but the question shows A as the intended answer format

Why A is correct:
Group the terms strategically: $(x^2y + xy^2) + (xz + yz)$. Factor out $xy$ from the first group and $z$ from the second: $xy(x + y) + z(x + y)$. Now $(x + y)$ is a common factor, giving $(x + y)(xy + z)$. ✓

Why the others are wrong:
- B: Expanding $(x + y)(xz + y)$ gives $x^2z + xy + xyz + y^2$—doesn't match the original.
- C: Expanding $(xy + z)(x - y)$ gives $x^2y - xy^2 + xz - yz$—wrong signs on middle terms.
- D: Expanding $(x + z)(y + z)$ gives $xy + xz + yz + z^2$—missing the $x^2y$ and $xy^2$ terms.

Why A is correct:
The greatest common factor (GCF) of all three terms is 3ab: 9a²b = 3ab·3a, 6ab² = 3ab·2b, and 3ab = 3ab·1. Factoring out 3ab gives 3ab(3a + 2b + 1). ✓

Why the others are wrong:
- B: Only factors out 3a, leaving b inside the bracket. When expanded: 3a(3ab + 2b² + b) = 9a²b + 6ab² + 3ab, but 3a is not the complete GCF—you can still pull out b.
- C: Only factors out 3b, not the complete GCF. Missing the factor of a.
- D: Doesn't factor out the GCF completely; 3ab is the largest common factor, not just ab. Also leaves 3 inside the bracket when it should be factored out.

Why A is correct:
First, find the greatest common factor (GCF) of all terms: 14m³, 21m², and 7m all share 7m. Factoring out 7m gives 7m(2m² − 3m + 1). Check: 7m × 2m² = 14m³ ✓, 7m × (−3m) = −21m² ✓, 7m × 1 = 7m ✓

Why the others are wrong:
- B: The signs are wrong inside the bracket—should be minus, not plus.
- C: This hasn't fully factored out the GCF; 7 is still hiding in each term inside the bracket.
- D: This hasn't factored out the m from 7m, leaving an incomplete factorisation.

A is correct:
- Find the GCF of all terms: $15x^2y^3$, $10xy^3$, and $5y^3$ share $5y^3$
- Factor it out: $5y^3(3x^2 - 2x + 1)$
- Check: $5y^3 \times 3x^2 = 15x^2y^3$ ✓, $5y^3 \times (-2x) = -10xy^3$ ✓, $5y^3 \times 1 = 5y^3$ ✓

Why others are wrong:
- B: Only factors out $5y^2$, leaving one factor of $y$ inside the bracket—incomplete factoring
- C: The middle term is wrong; should be $-2x$, not $+2x$
- D: Incorrectly factors out $5x$ instead of $5y^3$; doesn't match the original terms

A is correct: The greatest common factor of all three terms is 6pq. Dividing each term by 6pq gives: 12p²q ÷ 6pq = 2p, 18pq² ÷ 6pq = 3q, and 6pq ÷ 6pq = 1. So the factorization is 6pq(2p + 3q + 1).

B is wrong: It only factors out 6p, leaving q as a common factor inside the brackets that should have been factored out completely.

C is wrong: It only factors out 3q, leaving a common factor of 2p inside the brackets that wasn't fully extracted.

D is wrong: The terms inside the brackets are in the wrong order (2p + q + 3 instead of 2p + 3q + 1), which doesn't match the original expression when expanded.

Why A is correct:
First, find the greatest common factor (GCF) of all terms: 20u³, 30u², and 10u share a GCF of 10u. Factoring this out gives 10u(2u² − 3u + 1). You can verify by expanding: 10u × 2u² = 20u³, 10u × (−3u) = −30u², and 10u × 1 = 10u. ✓

Why the others are wrong:
- B: The signs are wrong inside the brackets (+ 3u instead of − 3u), so it doesn't match the original expression.
- C: The GCF is only 5u, not 10u—this doesn't fully factor out the common factors.
- D: The GCF is only 2u, not 10u—again, incomplete factoring that leaves common factors inside the brackets.

# Explanation

Why A is correct:
The greatest common factor of all three terms (8r²s, 12rs², 4rs) is 4rs. Factoring this out: 8r²s ÷ 4rs = 2r, 12rs² ÷ 4rs = 3s, and 4rs ÷ 4rs = 1. So the answer is 4rs(2r + 3s + 1). You can check by expanding: 4rs × 2r + 4rs × 3s + 4rs × 1 = 8r²s + 12rs² + 4rs ✓

Why B is wrong:
It only factors out 4r, not the full GCF of 4rs. The s in the last term (s) doesn't divide evenly into all original terms the same way.

Why C is wrong:
It only factors out 4s, missing the r. Also, the coefficients inside the brackets don't match what you'd get from dividing the original terms.

Why D is wrong:
While it correctly factors out 4rs, the terms inside are rearranged incorrectly: it has 2r + s + 3 instead of 2r + 3s + 1, which won't give the original expression when expanded.

Why A is correct:
The greatest common factor of all three terms is $7cd$. Factoring this out leaves $3c^2 - 2c + 1$. Check: $7cd(3c^2) = 21c^3d$ ✓, $7cd(-2c) = -14c^2d$ ✓, $7cd(1) = 7cd$ ✓

Why the others are wrong:
- B: Only factors out $7c$, leaving $d$ inside the bracket. This is incomplete factoring and doesn't match the original when expanded.
- C: Only factors out $7d$, missing the $c$ from the GCF. This also leaves $c$ terms in the bracket that shouldn't be there.
- D: Has a $+2c$ instead of $-2c$, and a $-1$ instead of $+1$—this doesn't match the original expression.

A is correct:
- The greatest common factor of all three terms is $8w^2$ (largest power of $w$ and largest number dividing 16, 24, and 8)
- Dividing each term by $8w^2$ gives: $16w^4 ÷ 8w^2 = 2w^2$, $24w^3 ÷ 8w^2 = 3w$, $8w^2 ÷ 8w^2 = 1$
- Result: $8w^2(2w^2 + 3w + 1)$ ✓

Why others are wrong:
- B: Only factors out $8w$, not the complete GCF; leaves an unfactored cubic inside
- C: The GCF is $8w^2$, not $4w^2$; this answer is incompletely factored
- D: Has a minus sign in the middle term, but the original has $+24w^3$ (positive), so this doesn't match

Why A is correct:
- The greatest common factor of all three terms is $6tu$ (since each term contains at least $6$, $t$, and $u$)
- Factoring out $6tu$ gives: $6tu(3t + 2u + 1)$ ✓

Why the others are wrong:
- B: Only factors out $6t$, leaving $u$ terms inside the bracket. This doesn't fully factor the expression.
- C: Only factors out $6u$, leaving $t$ terms inside. Again, incomplete factoring—$t$ appears in every term but isn't fully extracted.
- D: Has the right common factor $6tu$, but the bracket is wrong. If you expand $6tu(3t + u + 2)$, you get $18t^2u + 6tu^2 + 12tu$, which doesn't match the original.

A is correct.

The greatest common factor of all three terms (25v³, 15v², 5v) is 5v. Factoring this out: 25v³ ÷ 5v = 5v², 15v² ÷ 5v = 3v, and 5v ÷ 5v = 1, giving 5v(5v² − 3v + 1).

Why others are wrong:
- B: Has wrong signs (+ and −) inside the brackets—doesn't match the original expression when expanded.
- C: Uses 5v² as the common factor, which doesn't divide evenly into 5v (the last term).
- D: Only factors out v, not the complete GCF of 5v, so it's not fully factored.

Correct answer: (x + y)(x + z)

Expand this: (x + y)(x + z) = x² + xz + xy + yz ✓ This matches perfectly.

Why the others are wrong:
- B: (x + z)(y + z) expands to xy + xz + yz + z², which has z² instead of x²
- C: (x + y)(y + z) expands to xy + xz + y² + yz, which has y² instead of x²
- D: x(x + y + z) expands to x² + xy + xz, missing the yz term

Key strategy: Group the original expression as x(x + y) + z(x + y), then factor out the common bracket (x + y).

Why A is correct:
Group the terms: $(a^2b + ab^2) + (ac + bc)$
- First group: $ab(a + b)$
- Second group: $c(a + b)$
- Factor out $(a + b)$: $(a + b)(ab + c)$ ✓

Why the others are wrong:
- B: $(a + b)(ac + b)$ expands to $a^2c + ab + abc + b^2$—doesn't match the original
- C: $(a + b)(ab - c)$ has a minus sign, giving $a^2b + ab^2 - ac - bc$—wrong signs on the last two terms
- D: $(ab + c)(a - b)$ expands to $a^2b - ab^2 + ac - bc$—wrong middle term and wrong sign on $bc$

Why A is correct:
Group the terms: $(4x + 8y) + (2xz + 4yz) = 4(x + 2y) + 2z(x + 2y) = (x + 2y)(4 + 2z)$. Expanding confirms: $(x + 2y)(4 + 2z) = 4x + 2xz + 8y + 4yz$ ✓

Why the others are wrong:
- B: $(x + 2y)(2 + 4z) = 2x + 4xz + 4y + 8yz$ — coefficients don't match the original
- C: $(x + 4y)(2 + z) = 2x + xz + 8y + 4yz$ — missing the $4x$ term and has wrong coefficient on $8y$
- D: This doesn't properly factor; the first bracket is already a partially factored form, not a true common factor

Correct answer: A. $(m + 2n)(5 + 2p)$

• Group the terms strategically: $(5m + 10n) + (2mp + 4np)$
• Factor each group: $5(m + 2n) + 2p(m + 2n)$
• The common factor $(m + 2n)$ can be factored out: $(m + 2n)(5 + 2p)$ ✓

Why others are wrong:

• B: Expands to $5m + 2mn + 10p + 4np$ — doesn't match the original
• C: Expands to $2m + mp + 4n + 2np$ — missing the $10n$ term
• D: Expands to $5m + 4mp + 5n + 4np$ — the coefficients don't match

Why A is correct:
Group the terms as $(6p^2q + 3pq) + (8pr + 4r)$. Factor out $3pq$ from the first group and $4r$ from the second: $3pq(2p + 1) + 4r(2p + 1)$. Now $(2p + 1)$ is a common factor, giving $(2p + 1)(3pq + 4r)$. ✓

Why the others are wrong:
- B: $(p + 2)(3pq + 4r) = 3p^2q + 4pr + 6pq + 8r$ — doesn't match the original.
- C: $(2p + 1)(3p + 4qr) = 6p^2 + 8pqr + 3p + 4qr$ — wrong terms entirely.
- D: $(2p + 4r)(3pq + 1) = 6p^2q + 2p + 12pqr + 4r$ — has an extra $12pqr$ term.

Why A is correct:
Group the terms in pairs: $(9xy + 6y) + (3xz + 2z)$. Factor out $3y$ from the first pair and $z$ from the second: $3y(3x + 2) + z(3x + 2)$. Now $(3x + 2)$ is a common factor, giving $(3x + 2)(3y + z)$. ✓

Why the others are wrong:
- B: Expanding $(x + 2)(9y + z)$ gives $9xy + xz + 18y + 2z$ — doesn't match the original.
- C: Expanding $(3x + 2)(y + 3z)$ gives $3xy + 9xz + 2y + 6z$ — different terms.
- D: Expanding $(3x + z)(3y + 2)$ gives $9xy + 6x + 3yz + 2z$ — has $6x$ and $3yz$ instead of $6y$ and $3xz$.

Why A is correct:
Group the terms in pairs: $(7a^2 + 14a) + (3ab + 6b)$. Factor each group: $7a(a + 2) + 3b(a + 2)$. Now $(a + 2)$ is a common factor, so you get $(a + 2)(7a + 3b)$. ✓

Why the others are wrong:
- B: Expanding $(7a + 2)(a + 3b)$ gives $7a^2 + 21ab + 2a + 6b$ — the middle terms don't match the original.
- C: Expanding $(a + b)(7a + 6)$ gives $7a^2 + 6a + 7ab + 6b$ — again, wrong middle terms.
- D: The minus sign is wrong; expanding $(a + 2)(7a - 3b)$ gives $7a^2 - 3ab + 14a - 6b$, which doesn't match.

Correct answer: (2u + v)(4u + 3w)

Expand to check: $(2u + v)(4u + 3w) = 8u^2 + 6uw + 4uv + 3vw$ ✓ This matches perfectly.

Why the others are wrong:

• Option B: $(2u + w)(4u + 3v) = 8u^2 + 6uv + 4uw + 3vw$ — has $6uv$ instead of $4uv$, and $4uw$ instead of $6uw$.

• Option C: $(4u + v)(2u + 3w) = 8u^2 + 12uw + 2uv + 3vw$ — has wrong coefficients for the middle terms.

• Option D: $(2u + v)(2u + 3w) = 4u^2 + 6uw + 2uv + 3vw$ — only gives $4u^2$, not $8u^2$.

Correct answer: A

Expand $(2x + 1)(5xy + 1)$: $(2x)(5xy) + (2x)(1) + (1)(5xy) + (1)(1) = 10x^2y + 2x + 5xy + 1$ ✓

Why the others are wrong:

• B: $(5x + 1)(2xy + 1) = 10x^2y + 5x + 2xy + 1$ — gives $5x$ instead of $2x$
• C: $(2x + 1)(5x + y) = 10x^2 + 2xy + 5x + y$ — wrong powers and terms entirely
• D: $(10xy + 5)(x + 1) = 10x^2y + 10xy + 5x + 5$ — gives $10xy$ instead of $5xy$

Why A is correct:
Group the terms as $(p^2 - pq) + (pr - qr)$, then factor out common factors: $p(p - q) + r(p - q)$. Now $(p - q)$ is common to both terms, so you get $(p - q)(p + r)$.

Why the others are wrong:
- B & C: These are the same factorization written in different order, but if you expand $(p + q)(p - r)$, you get $p^2 - pr + pq - qr$, which doesn't match the original (the $pq$ term has the wrong sign).
- D: Expanding $(p - q)(r + q)$ gives $pr + pq - qr - q^2$, which has an extra $pq$ term and a $-q^2$ that aren't in the original.

Why A is correct:
Factorising means finding the greatest common factor (GCF) of both terms. The GCF of 12x² and 8x is 4x. When you divide each term by 4x, you get 3x + 2, so the answer is 4x(3x + 2). You can check: 4x × 3x + 4x × 2 = 12x² + 8x ✓

Why the others are wrong:
- B: 2x(6x + 8) only takes out 2x, not the full GCF. Also, you could factor 8 further from the bracket.
- C: The signs are wrong—there's no negative in the original expression.
- D: This leaves an x inside the bracket, so it hasn't been fully factorised.

Why A is correct:
- Find the greatest common factor (GCF) of the terms: $18a^3$ and $24a^2$ share $6a^2$
- Factor it out: $18a^3 - 24a^2 = 6a^2(3a - 4)$
- Check: $6a^2 \times 3a = 18a^3$ ✓ and $6a^2 \times (-4) = -24a^2$ ✓

Why others are wrong:
- B: Doesn't fully factor out common terms; $6a$ isn't the complete GCF
- C: Wrong sign—should be minus, not plus (doesn't match the original expression)
- D: $12a^2$ is not a common factor of both terms ($18a^3$ isn't divisible by 12)

Why A is correct:
The greatest common factor of $25m^2n$ and $15mn$ is $5mn$. Factoring this out: $25m^2n ÷ 5mn = 5m$ and $15mn ÷ 5mn = 3$, giving $5mn(5m + 3)$. ✓

Why the others are wrong:
- B: $5m(5mn + 3n)$ doesn't fully factor out the common factors—you can still pull out $n$ from inside the brackets.
- C: $5n(5m^2 + 3m)$ leaves out the $n$ from $25m^2n$, and you could factor out more $m$ from the brackets.
- D: $15mn$ isn't the GCF; the GCF is only $5mn$, so this doesn't give the correct factorization.

Why A is correct:
- Find the greatest common factor (GCF) of both terms: GCF of 14p³q² and 7p²q is 7p²q
- Factor it out: 7p²q(2pq − 1)
- Check: 7p²q × 2pq = 14p³q² ✓ and 7p²q × (−1) = −7p²q ✓

Why the others are wrong:
- B: Uses 7pq instead of 7p²q as the GCF—not the greatest common factor
- C: Has a plus sign instead of minus; doesn't match the original expression
- D: Uses 14p²q as the factor, which is too large and leaves pq − 1, giving the wrong expanded form

Why A is correct:
The greatest common factor of all three terms (30u²v, −20uv, 10v) is 10v. Factoring this out gives 10v(3u² − 2u + 1). You can verify by expanding: 10v × 3u² = 30u²v, 10v × (−2u) = −20uv, and 10v × 1 = 10v ✓

Why the others are wrong:
- B: Uses 10u as the common factor, but 10u doesn't divide into 10v (the last term)
- C: The signs are wrong—the middle term should be −2u, not +2u
- D: Uses 20v as the common factor, but the GCF is only 10v, not 20v

A is correct because:
- The GCF of $9k^2$ and $6k^3$ is $3k^2$
- Factoring out $3k^2$ gives: $3k^2(3 - 2k)$ ✓
- Check: $3k^2 \times 3 = 9k^2$ and $3k^2 \times (-2k) = -6k^3$ ✓

B is wrong: It only factors out $3k$, not the complete GCF. Also, $3k(3k - 2k^2)$ still has a common factor inside the brackets.

C is wrong: The GCF is $3k^2$, not $6k^2$. Also, $6k^2(3-k) = 18k^2 - 6k^3$, which doesn't match.

D is wrong: The signs are reversed—this gives $3k^2(2k - 3) = 6k^3 - 9k^2$, the opposite of what we need.

A is correct because:
- The GCF of all terms is $8t^2u$ (highest power of each common factor)
- Dividing each term by $8t^2u$ gives $2t^2 - 3tu + u^2$ ✓

B is wrong: Has $u^2$ in the GCF, but the original terms only have $u^1$ at minimum (first term is $16t^4u$).

C is wrong: The GCF is only $4t^2u$, not fully factored. Also, $4t^2u(4t^2 - 6tu + 2u^2)$ can still be simplified further by removing factor of 2.

D is wrong: The middle term should be $-3tu$ (negative), not $+3tu$ (positive).

A is correct:
- Find the GCF of both terms: $21r^3s^2$ and $14r^2s^2$ share $7r^2s^2$
- Factor it out: $7r^2s^2(3r + 2)$ ✓
- Check: $7r^2s^2 \times 3r = 21r^3s^2$ and $7r^2s^2 \times 2 = 14r^2s^2$ ✓

Why others are wrong:
- B: Uses $7rs^2$ instead of $7r^2s^2$—doesn't fully factor out the highest power of $r$
- C: The GCF is $7r^2s^2$, not $14r^2s^2$; also $(3r + 1)$ doesn't work when expanded
- D: Uses $7r^2s$ (missing an $s$) and has $(3r + 2s)$ which gives the wrong second term when expanded

A is correct: Find the greatest common factor (GCF) of all terms: 6z divides evenly into each term. Divide each term by 6z: $12z^3 ÷ 6z = 2z^2$, $18z^2 ÷ 6z = 3z$, $6z ÷ 6z = 1$. This gives $6z(2z^2 + 3z + 1)$.

B is wrong: The terms inside the bracket are rearranged incorrectly—it has $2z^2 + z + 3$ instead of $2z^2 + 3z + 1$.

C is wrong: While 3z is a common factor, it's not the *greatest* common factor. Using it leaves common factors still inside the bracket ($6z$ is still hidden).

D is wrong: The GCF is 6z, not $6z^2$ (since the last term is just $6z$, not $6z^2$). Also, the bracket notation is incorrect (you can't write $2z + 3 + 1$ this way).

Why A is correct:
Factor out the greatest common factor (GCF) of all terms: $4y^2$ divides evenly into $8y^4$, $12y^3$, and $4y^2$. This leaves $2y^2 - 3y - 1$ inside the brackets. Expanding $4y^2(2y^2 - 3y - 1)$ gives back the original expression.

Why the others are wrong:
- B: The GCF is $4y^2$, not just $4y$. Expanding this gives $8y^4 - 12y^3 - 4y^2$... wait, let me check: $4y(2y^3) = 8y^4$ ✓, but this hasn't fully factored out the common factor.
- C: Hasn't factored out the largest common factor; $2y^2$ is smaller than $4y^2$. Also, $2y^2(4y^2 - 6y - 2)$ can be simplified further.
- D: The signs are wrong inside the brackets—should be minus signs, not plus signs. This would give $8y^4 + 12y^3 + 4y^2$, not the original expression.

Why A is correct:
Group the terms: $(x^2y - xy^2) + (xz - yz)$. Factor each group: $xy(x - y) + z(x - y)$. Now $(x - y)$ is a common factor, giving $(x - y)(xy + z)$. ✓

Why others are wrong:
- B: Expanding $(x + y)(xy - z)$ gives $x^2y - xz + xy^2 - yz$, which doesn't match (wrong signs).
- C: Expanding $(x - y)(x + z)$ gives $x^2 + xz - xy - yz$—missing the $y^2$ term entirely.
- D: Expanding $(xy + z)(x + y)$ gives $x^2y + xy^2 + xz + yz$ (wrong signs on the last two terms).