https://file.aiquickdraw.com/mathbot/file/17262656062773rwwvtxg.jpg

Let's solve the two questions in the image step by step.

### **Question 1**: Find factors of \( x^3 - 7x - 6 \)

We want to factor \( x^3 - 7x - 6 \). To do this, let's first attempt synthetic division or trial and error using rational roots.

#### Step 1: Check Rational Roots
Possible rational roots are factors of \(-6\) (constant term) over factors of 1 (leading coefficient of \(x^3\)), i.e., \( \pm 1, \pm 2, \pm 3, \pm 6\).

We'll try substituting these values into the equation to check if they are roots.

- Try \(x = -1\):
  \[
  (-1)^3 - 7(-1) - 6 = -1 + 7 - 6 = 0
  \]
  So, \(x = -1\) is a root.

#### Step 2: Polynomial Division
Since \(x = -1\) is a root, one factor is \(x + 1\). Now, divide \(x^3 - 7x - 6\) by \(x + 1\) using synthetic or polynomial division:

Dividing \(x^3 - 7x - 6\) by \(x + 1\), we get \(x^2 - x - 6\).

#### Step 3: Factor the Quadratic
Now, factor \(x^2 - x - 6\):
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

So, the full factorization is:
\[
x^3 - 7x - 6 = (x + 1)(x - 3)(x + 2)
\]

The correct option for **Question 1** is **C: \( (x - 3)(x + 2)(x + 1) \)**.

---

### **Question 2**: Find factors of \( x^3 + 12x^2 - 19x = -20 \)

First, rewrite the equation as:
\[
x^3 + 12x^2 - 19x + 20 = 0
\]

#### Step 1: Check Rational Roots
The possible rational roots are factors of \(20\) over factors of \(1\), i.e., \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \).

- Try \(x = 1\):
  \[
  1^3 + 12(1)^2 - 19(1) + 20 = 1 + 12 - 19 + 20 = 14 \quad (\text{not a root})
  \]

- Try \(x = -1\):
  \[
  (-1)^3 + 12(-1)^2 - 19(-1) + 20 = -1 + 12 + 19 + 20 = 50 \quad (\text{not a root})
  \]

- Try \(x = 5\):
  \[
  5^3 + 12(5)^2 - 19(5) + 20 = 125 + 300 - 95 + 20 = 350 \quad (\text{not a root})
  \]

- Try \(x = -5\):
  \[
  (-5)^3 + 12(-5)^2 - 19(-5) + 20 = -125 + 300 + 95 + 20 = 290 \quad (\text{wrong})

Find factors of x^3 - 7x - 6 and x^3 + 12x^2 - 19x = -20

Learn how to solve cubic polynomial factorization problems, such as finding the factors of x^3 - 7x - 6 and x^3 + 12x^2 - 19x = -20, using methods like synthetic division and the Rational Root Theorem. This step-by-step solution is suitable for students in Grades 9-12.

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