9. factor $6r^2 - 3rs - 3s^2$
10. factor $6x^3 - x^2 - 12x$
11. factor $8m^3n^2 - 32m^2n^3 + 24mn^4$
12. factor $30x^2 + 3xy - 6y^2$
13. factor $2x^3 + 5x^2 - 3x$
14. factor $m^4 - 16$
15. factor $x^2 + 4xy + 4y^2 - 4z^2$
16. factor $m^4 - m^2 - 12$
17. factor $(2x + 3)^2 + 2(2x + 3) - 15$

Question

Accepted Answer

### Problem 9: Factor $6r^2 - 3rs - 3s^2$
# Explanation:
## Step 1: Factor out the GCF
The greatest common factor (GCF) of $6r^2$, $-3rs$, and $-3s^2$ is $3$.
$6r^2 - 3rs - 3s^2 = 3(2r^2 - rs - s^2)$

## Step 2: Factor the quadratic
Factor $2r^2 - rs - s^2$. We need two numbers that multiply to $2\times(-1)= -2$ and add to $-1$. The numbers are $-2$ and $1$.
Rewrite the middle term: $2r^2 - 2rs + rs - s^2$
Group and factor:
$= 2r(r - s) + s(r - s)$
$= (2r + s)(r - s)$

## Step 3: Combine the factors
Multiply the GCF back in: $3(2r + s)(r - s)$

# Answer:
$3(2r + s)(r - s)$

### Problem 10: Factor $6x^3 - x^2 - 12x$
# Explanation:
## Step 1: Factor out the GCF
The GCF of $6x^3$, $-x^2$, and $-12x$ is $x$.
$6x^3 - x^2 - 12x = x(6x^2 - x - 12)$

## Step 2: Factor the quadratic
Factor $6x^2 - x - 12$. We need two numbers that multiply to $6\times(-12)= -72$ and add to $-1$. The numbers are $-9$ and $8$.
Rewrite the middle term: $6x^2 + 8x - 9x - 12$
Group and factor:
$= 2x(3x + 4) - 3(3x + 4)$
$= (2x - 3)(3x + 4)$

## Step 3: Combine the factors
$x(2x - 3)(3x + 4)$

# Answer:
$x(2x - 3)(3x + 4)$

### Problem 11: Factor $8m^3n^2 - 32m^2n^3 + 24mn^4$
# Explanation:
## Step 1: Factor out the GCF
The GCF of $8m^3n^2$, $-32m^2n^3$, and $24mn^4$ is $8mn^2$.
$8m^3n^2 - 32m^2n^3 + 24mn^4 = 8mn^2(m^2 - 4mn + 3n^2)$

## Step 2: Factor the quadratic
Factor $m^2 - 4mn + 3n^2$. We need two numbers that multiply to $3$ and add to $-4$. The numbers are $-3$ and $-1$.
$m^2 - 4mn + 3n^2 = (m - 3n)(m - n)$

## Step 3: Combine the factors
$8mn^2(m - 3n)(m - n)$

# Answer:
$8mn^2(m - 3n)(m - n)$

### Problem 12: Factor $30x^2 + 3xy - 6y^2$
# Explanation:
## Step 1: Factor out the GCF
The GCF of $30x^2$, $3xy$, and $-6y^2$ is $3$.
$30x^2 + 3xy - 6y^2 = 3(10x^2 + xy - 2y^2)$

## Step 2: Factor the quadratic
Factor $10x^2 + xy - 2y^2$. We need two numbers that multiply to $10\times(-2)= -20$ and add to $1$. The numbers are $5$ and $-4$.
Rewrite the middle term: $10x^2 + 5xy - 4xy - 2y^2$
Group and factor:
$= 5x(2x + y) - 2y(2x + y)$
$= (5x - 2y)(2x + y)$

## Step 3: Combine the factors
$3(5x - 2y)(2x + y)$

# Answer:
$3(5x - 2y)(2x + y)$

### Problem 13: Factor $2x^3 + 5x^2 - 3x$
# Explanation:
## Step 1: Factor out the GCF
The GCF of $2x^3$, $5x^2$, and $-3x$ is $x$.
$2x^3 + 5x^2 - 3x = x(2x^2 + 5x - 3)$

## Step 2: Factor the quadratic
Factor $2x^2 + 5x - 3$. We need two numbers that multiply to $2\times(-3)= -6$ and add to $5$. The numbers are $6$ and $-1$.
Rewrite the middle term: $2x^2 + 6x - x - 3$
Group and factor:
$= 2x(x + 3) - 1(x + 3)$
$= (2x - 1)(x + 3)$

## Step 3: Combine the factors
$x(2x - 1)(x + 3)$

# Answer:
$x(2x - 1)(x + 3)$

### Problem 14: Factor $m^4 - 16$
# Explanation:
## Step 1: Recognize as a difference of squares
$m^4 - 16 = (m^2)^2 - 4^2 = (m^2 - 4)(m^2 + 4)$

## Step 2: Factor $m^2 - 4$ (another difference of squares)
$m^2 - 4 = m^2 - 2^2 = (m - 2)(m + 2)$

## Step 3: Combine the factors
$(m - 2)(m + 2)(m^2 + 4)$

# Answer:
$(m - 2)(m + 2)(m^2 + 4)$

### Problem 15: Factor $x^2 + 4xy + 4y^2 - 4z^2$
# Explanation:
## Step 1: Factor the perfect square trinomial
$x^2 + 4xy + 4y^2 = (x + 2y)^2$, so the expression becomes $(x + 2y)^2 - 4z^2$

## Step 2: Recognize as a difference of squares
$(x + 2y)^2 - (2z)^2 = (x + 2y - 2z)(x + 2y + 2z)$

# Answer:
$(x + 2y - 2z)(x + 2y + 2z)$

### Problem 16: Factor $m^4 - m^2 - 12$
# Explanation:
## Step 1: Let $u = m^2$ (substitution)
The expression becomes $u^2 - u - 12$

## Step 2: Factor the quadratic in $u$
We need two numbers that multiply to $-12$ and add to $-1$. The numbers are $-4$ and $3$.
$u^2 - u - 12 = (u - 4)(u + 3)$

## Step 3: Substitute back $u = m^2$
$(m^2 - 4)(m^2 + 3)$

## Step 4: Factor $m^2 - 4$ (difference of squares)
$m^2 - 4 = (m - 2)(m + 2)$

## Step 5: Combine the factors
$(m - 2)(m + 2)(m^2 + 3)$

# Answer:
$(m - 2)(m + 2)(m^2 + 3)$

### Problem 17: Factor $(2x + 3)^2 + 2(2x + 3) - 15$
# Explanation:
## Step 1: Let $u = 2x + 3$ (substitution)
The expression becomes $u^2 + 2u - 15$

## Step 2: Factor the quadratic in $u$
We need two numbers that multiply to $-15$ and add to $2$. The numbers are $5$ and $-3$.
$u^2 + 2u - 15 = (u + 5)(u - 3)$

## Step 3: Substitute back $u = 2x + 3$
$(2x + 3 + 5)(2x + 3 - 3) = (2x + 8)(2x)$

## Step 4: Simplify the factors
Factor out $2$ from $2x + 8$: $2(x + 4)\times2x = 4x(x + 4)$ (or we can also factor as $2x(2x + 8) = 2x\times2(x + 4)=4x(x + 4)$)

# Answer:
$4x(x + 4)$ (or $2x(2x + 8)$ or $(2x + 8)(2x)$, but simplified form is $4x(x + 4)$)

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QUESTION IMAGE

Question

Explanation:

Problem 9: Factor \(6r^2 - 3rs - 3s^2\)

Step 1: Factor out the GCF

Step 2: Factor the quadratic

Step 3: Combine the factors

Step 1: Factor out the GCF

Step 2: Factor the quadratic

Step 3: Combine the factors

Step 1: Factor out the GCF

Step 2: Factor the quadratic

Step 3: Combine the factors

Answer:

Problem 10: Factor \(6x^3 - x^2 - 12x\)