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### Given Polynomial
\[
P(x) = x^3 + 5x^2 + 4x + 20
\]

### Step 1: Find the Zeros
Using the Rational Root Theorem and synthetic division, we determined that the roots (zeros) of the polynomial are:
- \( x = -2 \)
- \( x = -4 \)
- \( x = 1 \)

### Step 2: Factored Form
The polynomial \( P(x) \) can be written in its fully factored form as:
\[
P(x) = (x + 2)(x + 4)(x - 1)
\]

### Final Answer
- **The zeros are:** \( -2, -4, \) and \( 1 \)
- **The factored form of** \( P(x) \) **is:**
  \[
  P(x) = (x + 2)(x + 4)(x - 1)
  \]

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### Related Questions
1. How do you determine if a root is a repeated root?
2. Can a polynomial of degree 3 have more than three real roots?
3. How does synthetic division help in finding polynomial roots?
4. What happens if none of the rational roots work—how do we proceed then?
5. How can you verify that your factored form is correct?

### Tip
Always substitute the roots back into the original polynomial to verify that they make the equation equal to zero, ensuring your solutions are correct.

Given P(x) = x^3 + 5x^2 + 4x + 20, find the zeros, real and non-real of P, and write P in factored form.

Learn how to find the real and non-real zeros of the polynomial P(x) = x^3 + 5x^2 + 4x + 20. This guide covers synthetic division and the Rational Root Theorem, providing a step-by-step explanation for factoring polynomials. Suitable for high school students in Grades 10-12.

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