To find the zeros of the polynomial function:

\[
f(x) = x^4 + 6x^3 + 9x^2 + 24x + 20
\]

We'll use the **Rational Root Theorem** to test possible rational roots, followed by synthetic division or factorization.

### Step 1: Apply the Rational Root Theorem
According to 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, in this case). The possible rational roots are:

\[
\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20
\]

### Step 2: Test possible roots
We can test these possible values using substitution into the polynomial to see if they are indeed roots.

- For \(x = -2\):

\[
f(-2) = (-2)^4 + 6(-2)^3 + 9(-2)^2 + 24(-2) + 20 = 16 - 48 + 36 - 48 + 20 = 0
\]

So, \(x = -2\) is a root.

### Step 3: Polynomial division
Now that we know \(x = -2\) is a root, we can divide \(f(x)\) by \(x + 2\).

Using synthetic division, we divide \(x^4 + 6x^3 + 9x^2 + 24x + 20\) by \(x + 2\):

\[
x^4 + 6x^3 + 9x^2 + 24x + 20 \div (x + 2) = x^3 + 4x^2 + x + 10
\]

### Step 4: Solve the quotient polynomial
We now have a cubic equation:

\[
x^3 + 4x^2 + x + 10 = 0
\]

Again, we test possible rational roots. Testing \(x = -2\) again:

\[
(-2)^3 + 4(-2)^2 + (-2) + 10 = -8 + 16 - 2 + 10 = 16
\]

So, \(x = -2\) is not a root of the cubic. Testing other values doesn't yield simple results, so we resort to numerical or further factorization methods (e.g., Cardano's method) to solve the cubic equation.

Thus, the exact rational root is \(x = -2\), and the cubic part can be solved numerically or by more advanced methods for exact roots.

---

Would you like more details on solving the cubic equation? 

### Follow-up Questions:
1. How does the Rational Root Theorem work in finding possible roots?
2. What is synthetic division, and how is it applied?
3. Could there be any complex roots for this cubic polynomial?
4. How can you solve cubic equations algebraically or numerically?
5. Can polynomial division always simplify finding roots?

### Tip:
If a polynomial root cannot be found easily using rational methods, graphical or numerical techniques like Newton's method are often helpful.

Learn how to find the zeros of the polynomial equation x^4 + 6x^3 + 9x^2 + 24x + 20 using the Rational Root Theorem and synthetic division. This solution covers polynomial factorization and explores methods for solving cubic equations. Suitable for high school students in Grades 10-12.

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