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The problem you uploaded is asking to find the zeros of the cubic function \( h(x) = x^3 - 5x^2 + 2x + 12 \).

### Steps to find the zeros:
We need to solve the equation:
\[
x^3 - 5x^2 + 2x + 12 = 0
\]
Here’s a structured way to approach it:

1. **Use the Rational Root Theorem**: The possible rational roots are the factors of the constant term (12) divided by the factors of the leading coefficient (1). The possible rational roots are:
   \[
   \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
   \]

2. **Test possible roots**: We can substitute these values into the equation to see which one satisfies the equation. Let's test \(x = 2\):
   \[
   h(2) = (2)^3 - 5(2)^2 + 2(2) + 12 = 8 - 20 + 4 + 12 = 4
   \]
   So, \(x = 2\) is not a root. Let’s try \(x = -1\):
   \[
   h(-1) = (-1)^3 - 5(-1)^2 + 2(-1) + 12 = -1 - 5 - 2 + 12 = 4
   \]
   This also isn’t a root. Continue testing until a root is found, or use synthetic division for quicker calculation.

3. **Factorization or synthetic division**: After finding one root, divide the cubic polynomial by \((x - \text{root})\), and then solve the resulting quadratic equation for the remaining roots.

Would you like more details on synthetic division or further explanation on testing roots?

### Expansion Questions:
1. How does the Rational Root Theorem help in solving polynomial equations?
2. Why is it important to factor a cubic equation after finding a root?
3. What is synthetic division, and how does it simplify the process?
4. Can a cubic function have more than three real roots? Why or why not?
5. How can the Descartes’ Rule of Signs help predict the number of real roots?

### Tip:
Testing small integers (positive and negative) first can often quickly lead you to a rational root when solving cubic equations manually.

Find all the zeros of h(x) = x^3 - 5x^2 + 2x + 12. Write the answer in exact form. If there is more than one answer, separate them with commas. Select 'None' if applicable.

Explore how to find the exact zeros of the cubic polynomial h(x) = x^3 - 5x^2 + 2x + 12. This solution walks through the Rational Root Theorem, testing potential roots, and synthetic division, making it ideal for high school algebra students (Grades 9-12).

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