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To factor the cubic polynomial \( x^3 + 5x^2 + 4x + 20 \) completely, let's proceed with the following steps:

### Step 1: Identify possible rational roots.
Using the Rational Root Theorem, the possible rational roots are the factors of the constant term (20) divided by the factors of the leading coefficient (1). This gives possible roots: \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \).

### Step 2: Test the possible roots.
We substitute these possible roots into the polynomial to check which one results in zero.

- \( x = -2 \):
  \[
  (-2)^3 + 5(-2)^2 + 4(-2) + 20 = -8 + 20 - 8 + 20 = 24 \quad (\text{not a root})
  \]

- \( x = -5 \):
  \[
  (-5)^3 + 5(-5)^2 + 4(-5) + 20 = -125 + 125 - 20 + 20 = 0 \quad (\text{root found})
  \]

Since \( x = -5 \) is a root, \( x + 5 \) is a factor.

### Step 3: Perform synthetic or polynomial division.
We divide the polynomial \( x^3 + 5x^2 + 4x + 20 \) by \( x + 5 \) to find the quotient.

Performing the division:

\[
\text{Quotient} = x^2 + 4
\]

### Step 4: Factor the remaining quadratic if possible.
The quotient \( x^2 + 4 \) cannot be factored further over the real numbers, as it has no real roots. It can be factored as:

\[
x^2 + 4 = (x + 2i)(x - 2i)
\]

### Final Factorization:
Thus, the complete factorization of the polynomial over the complex numbers is:

\[
x^3 + 5x^2 + 4x + 20 = (x + 5)(x + 2i)(x - 2i)
\]

Over the real numbers, the factorization would be:

\[
x^3 + 5x^2 + 4x + 20 = (x + 5)(x^2 + 4)
\]

Would you like more details on any of these steps, or do you have any specific questions?

Here are some related questions:
1. How does the Rational Root Theorem work?
2. What are complex roots and how do they affect polynomial factorization?
3. How do you perform synthetic division?
4. Can all cubic polynomials be factored this way?
5. What is the difference between factoring over real numbers and complex numbers?

**Tip:** Always test the simpler roots first (like \( \pm 1 \)) to quickly identify factors and reduce the polynomial.

Learn how to factor the cubic polynomial x^3 + 5x^2 + 4x + 20 completely using the Rational Root Theorem and complex numbers. This problem covers key concepts in polynomials, including factorization and complex roots. Suitable for high school students.

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