https://file.aiquickdraw.com/mathbot/file/1726569791734vh2obm5m.jpg

The expression provided in the image is:

\[
P(x) = y^3 + 5y^2 - 12y - 36
\]
where \(-2\) is given as one of the zeros of the polynomial.

To solve this problem, we will:

1. Use synthetic or polynomial division to divide the polynomial \(P(x)\) by \((y + 2)\) since \(-2\) is one of the roots.
2. Factor the resulting polynomial further if possible to find the remaining zeros.

Let's proceed by performing polynomial division.

### Synthetic Division Steps:
We will divide \(P(y)\) by \((y + 2)\) using synthetic division.

I'll perform the division now.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the steps of solving the problem manually if you'd like! Just let me know.

Would you like me to walk you through the synthetic division and next steps? 

Here are some related questions to expand on this:

1. How is synthetic division performed with polynomials?
2. What is the relationship between a polynomial's degree and the number of roots?
3. What does it mean when a number is a "zero" of a polynomial?
4. How do we factor the quotient polynomial after division?
5. Can we use the Rational Root Theorem to check for possible other roots?

**Tip**: When factoring cubic polynomials, using the known zeros helps reduce the degree of the polynomial for easier solutions.

P(x) = y^3 + 5y^2 - 12y - 36, given -2 as one of its zeros.

This problem involves solving the polynomial y^3 + 5y^2 - 12y - 36 given that -2 is one of its zeros. Using synthetic division, we find the remaining factors and explore the steps involved in determining all the roots of the cubic polynomial. Suitable for students in Grades 9-12.

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