Question

1. $12p^2 - 7p - 10$
2. $15d^2 + 16d - 15$
3. $2x^2 + 17x - 9$
4. $-9v^2 - 22v - 8$
5. $-14m^2 + 13m - 3$
6. $-20q^2 + 56q - 15$

Explanation:

Problem 7: $12p^2 -7p -10$

Step1: Split middle term

Find two numbers: product $12\times(-10)=-120$, sum $-7$. The numbers are $-15$ and $8$.
$12p^2 -15p +8p -10$

Step2: Factor by grouping

$3p(4p -5) +2(4p -5)$

Step3: Extract common factor

$(3p +2)(4p -5)$

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Problem 8: $15d^2 +16d -15$

Step1: Split middle term

Find two numbers: product $15\times(-15)=-225$, sum $16$. The numbers are $25$ and $-9$.
$15d^2 +25d -9d -15$

Step2: Factor by grouping

$5d(3d +5) -3(3d +5)$

Step3: Extract common factor

$(5d -3)(3d +5)$

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Problem 9: $2x^2 +17x -9$

Step1: Split middle term

Find two numbers: product $2\times(-9)=-18$, sum $17$. The numbers are $18$ and $-1$.
$2x^2 +18x -x -9$

Step2: Factor by grouping

$2x(x +9) -1(x +9)$

Step3: Extract common factor

$(2x -1)(x +9)$

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Problem 10: $-9v^2 -22v -8$

Step1: Factor out negative sign

$-(9v^2 +22v +8)$

Step2: Split middle term

Find two numbers: product $9\times8=72$, sum $22$. The numbers are $18$ and $4$.
$-(9v^2 +18v +4v +8)$

Step3: Factor by grouping

$-(9v(v +2) +4(v +2))$

Step4: Extract common factor

$-(9v +4)(v +2)$

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Problem 11: $-14m^2 +13m -3$

Step1: Factor out negative sign

$-(14m^2 -13m +3)$

Step2: Split middle term

Find two numbers: product $14\times3=42$, sum $-13$. The numbers are $-7$ and $-6$.
$-(14m^2 -7m -6m +3)$

Step3: Factor by grouping

$-(7m(2m -1) -3(2m -1))$

Step4: Extract common factor

$-(7m -3)(2m -1)$

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Problem 12: $-20q^2 +56q -15$

Step1: Factor out negative sign

$-(20q^2 -56q +15)$

Step2: Split middle term

Find two numbers: product $20\times15=300$, sum $-56$. The numbers are $-50$ and $-6$.
$-(20q^2 -50q -6q +15)$

Step3: Factor by grouping

$-(10q(2q -5) -3(2q -5))$

Step4: Extract common factor

$-(10q -3)(2q -5)$

Answer:

1. $\boldsymbol{(3p + 2)(4p - 5)}$
2. $\boldsymbol{(5d - 3)(3d + 5)}$
3. $\boldsymbol{(2x - 1)(x + 9)}$
4. $\boldsymbol{-(9v + 4)(v + 2)}$
5. $\boldsymbol{-(7m - 3)(2m - 1)}$
6. $\boldsymbol{-(10q - 3)(2q - 5)}$