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I can see the math problem from your screenshot, which asks to factor \( f(x) = 3x^3 + 28x^2 - x - 90 \) given that \( x = -2 \) is a zero.

To solve this, we can use synthetic division since we know that \( x = -2 \) is a root. Let's go through the steps:

### Step 1: Set up synthetic division
We use \( x + 2 \) as the divisor because the given zero is \( x = -2 \).

The coefficients of the polynomial \( f(x) = 3x^3 + 28x^2 - x - 90 \) are: \( [3, 28, -1, -90] \).

### Step 2: Perform synthetic division
We divide the polynomial by \( x + 2 \):

1. Bring down the first coefficient, \( 3 \).
2. Multiply \( 3 \) by \( -2 \), giving \( -6 \), and add to \( 28 \), resulting in \( 22 \).
3. Multiply \( 22 \) by \( -2 \), giving \( -44 \), and add to \( -1 \), resulting in \( -45 \).
4. Multiply \( -45 \) by \( -2 \), giving \( 90 \), and add to \( -90 \), resulting in \( 0 \) (remainder).

The quotient is \( 3x^2 + 22x - 45 \), with a remainder of 0, confirming that \( x = -2 \) is a factor.

### Step 3: Factor the quadratic
Now, we factor the quadratic \( 3x^2 + 22x - 45 \).

We need two numbers that multiply to \( 3 \times -45 = -135 \) and add to \( 22 \). These numbers are \( 27 \) and \( -5 \).

So, rewrite the middle term:
\[
3x^2 + 27x - 5x - 45
\]
Factor by grouping:
\[
3x(x + 9) - 5(x + 9)
\]
Factor out the common binomial:
\[
(3x - 5)(x + 9)
\]

### Final Answer:
The complete factorization of \( f(x) \) is:
\[
f(x) = (x + 2)(3x - 5)(x + 9)
\]

Would you like further details on any step? Here are five related questions to explore further:

1. How do you perform synthetic division for higher-degree polynomials?
2. What is the significance of the remainder in synthetic division?
3. How can we use the Rational Root Theorem to find potential roots?
4. What are the methods for factoring quadratic expressions?
5. How can you verify your factorization of a polynomial?

**Tip:** When factoring polynomials, always check your work by multiplying the factors back together to ensure you recover the original polynomial.

Factor f(x) = 3x^3 + 28x^2 - x - 90, given that x = -2 is a zero.

Learn how to factor the cubic polynomial f(x) = 3x^3 + 28x^2 - x - 90 given x = -2 as a zero. This problem involves synthetic division and quadratic factorization. Suitable for students in Grades 11-12 or early college courses.

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