factor each trinomial.
1. $6c^2 + 11c + 4$
2. $4x^2 + 2x - 12$
3. $x^2 - 8x + 15$
4. $x^2 - 3x - 40$
5. $2m^2 + m - 28$
6. $x^2 - 18x + 81$
7. $9x^2 + 48x + 64$

Question

Accepted Answer

### Problem 1: Factor $6c^2 + 11c + 4$
# Explanation:
## Step1: Find two numbers that multiply to $6\times4 = 24$ and add to $11$. The numbers are $8$ and $3$.
Rewrite the middle term: $6c^2 + 8c + 3c + 4$
## Step2: Group the terms.
$(6c^2 + 8c) + (3c + 4)$
## Step3: Factor out the GCF from each group.
$2c(3c + 4) + 1(3c + 4)$
## Step4: Factor out the common binomial factor.
$(2c + 1)(3c + 4)$
# Answer: $(2c + 1)(3c + 4)$

### Problem 2: Factor $4x^2 + 2x - 12$
# Explanation:
## Step1: Factor out the GCF of $2$ from all terms.
$2(2x^2 + x - 6)$
## Step2: Factor the quadratic inside the parentheses. Find two numbers that multiply to $2\times(-6)= -12$ and add to $1$. The numbers are $4$ and $-3$.
Rewrite the middle term: $2(2x^2 + 4x - 3x - 6)$
## Step3: Group the terms.
$2[(2x^2 + 4x) + (-3x - 6)]$
## Step4: Factor out the GCF from each group.
$2[2x(x + 2) - 3(x + 2)]$
## Step5: Factor out the common binomial factor.
$2(2x - 3)(x + 2)$
# Answer: $2(2x - 3)(x + 2)$

### Problem 3: Factor $x^2 - 8x + 15$
# Explanation:
## Step1: Find two numbers that multiply to $15$ and add to $-8$. The numbers are $-5$ and $-3$.
## Step2: Write the factored form.
$(x - 5)(x - 3)$
# Answer: $(x - 5)(x - 3)$

### Problem 4: Factor $x^2 - 3x - 40$
# Explanation:
## Step1: Find two numbers that multiply to $-40$ and add to $-3$. The numbers are $-8$ and $5$.
## Step2: Write the factored form.
$(x - 8)(x + 5)$
# Answer: $(x - 8)(x + 5)$

### Problem 5: Factor $2m^2 + m - 28$
# Explanation:
## Step1: Find two numbers that multiply to $2\times(-28)= -56$ and add to $1$. The numbers are $8$ and $-7$.
Rewrite the middle term: $2m^2 + 8m - 7m - 28$
## Step2: Group the terms.
$(2m^2 + 8m) + (-7m - 28)$
## Step3: Factor out the GCF from each group.
$2m(m + 4) - 7(m + 4)$
## Step4: Factor out the common binomial factor.
$(2m - 7)(m + 4)$
# Answer: $(2m - 7)(m + 4)$

### Problem 6: Factor $x^2 - 18x + 81$
# Explanation:
## Step1: Recognize this as a perfect square trinomial. $a^2 - 2ab + b^2=(a - b)^2$, where $a = x$ and $2ab = 18x$, so $b = 9$.
## Step2: Write the factored form.
$(x - 9)^2$
# Answer: $(x - 9)^2$

### Problem 7: Factor $9x^2 + 48x + 64$
# Explanation:
## Step1: Recognize this as a perfect square trinomial. $a^2 + 2ab + b^2=(a + b)^2$, where $a = 3x$ and $2ab = 48x$, so $b = 8$ (since $2\times3x\times8 = 48x$ and $8^2 = 64$).
## Step2: Write the factored form.
$(3x + 8)^2$
# Answer: $(3x + 8)^2$

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QUESTION IMAGE

Question

Explanation:

Problem 1: Factor \(6c^2 + 11c + 4\)

Step1: Find two numbers that multiply to \(6\times4 = 24\) and add to \(11\). The numbers are \(8\) and \(3\).

Step2: Group the terms.

Step3: Factor out the GCF from each group.

Step4: Factor out the common binomial factor.

Step1: Factor out the GCF of \(2\) from all terms.

Step2: Factor the quadratic inside the parentheses. Find two numbers that multiply to \(2\times(-6)= -12\) and add to \(1\). The numbers are \(4\) and \(-3\).

Step3: Group the terms.

Step4: Factor out the GCF from each group.

Step5: Factor out the common binomial factor.

Step1: Find two numbers that multiply to \(15\) and add to \(-8\). The numbers are \(-5\) and \(-3\).

Step2: Write the factored form.

Answer:

Problem 2: Factor \(4x^2 + 2x - 12\)