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Let's solve the polynomial function \( f(x) = x^3 + 3x^2 - 2x - 4 \).

### Step 1: Factor the Polynomial
We start by finding possible rational roots using the Rational Root Theorem. According to this theorem, possible roots are the factors of the constant term divided by the factors of the leading coefficient.

The constant term is \(-4\), and the leading coefficient is \(1\). So the possible rational roots are:

\[
\pm 1, \pm 2, \pm 4
\]

### Step 2: Test Possible Roots
Let's test these possible roots by substituting them into the polynomial.

1. **Test \( x = 1 \):**
   \[
   f(1) = (1)^3 + 3(1)^2 - 2(1) - 4 = 1 + 3 - 2 - 4 = -2
   \]
   So, \( x = 1 \) is not a root.

2. **Test \( x = -1 \):**
   \[
   f(-1) = (-1)^3 + 3(-1)^2 - 2(-1) - 4 = -1 + 3 + 2 - 4 = 0
   \]
   So, \( x = -1 \) is a root.

### Step 3: Factor the Polynomial Using \( x + 1 \)
Since \( x = -1 \) is a root, \( x + 1 \) is a factor. We can perform polynomial division to divide \( x^3 + 3x^2 - 2x - 4 \) by \( x + 1 \).

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

So, we can write:

\[
f(x) = (x + 1)(x^2 + 2x - 4)
\]

### Step 4: Solve the Quadratic Equation
Now, we solve the quadratic equation \( x^2 + 2x - 4 = 0 \).

The solutions are given by the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 2 \), and \( c = -4 \).

\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-4)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 16}}{2} = \frac{-2 \pm \sqrt{20}}{2} = \frac{-2 \pm 2\sqrt{5}}{2} = -1 \pm \sqrt{5}
\]

So the roots of the quadratic are:

\[
x = -1 + \sqrt{5} \quad \text{and} \quad x = -1 - \sqrt{5}
\]

### Final Solution
The solutions to the polynomial equation \( f(x) = 0 \) are:

\[
x = -1, \quad x = -1 + \sqrt{5}, \quad x = -1 - \sqrt{5}
\]

Would you like more details or have any questions?

Here are five related questions you might find useful:
1. How do you perform polynomial division manually?
2. What are the properties of cubic polynomials?
3. How can the Rational Root Theorem be applied to higher-degree polynomials?
4. What is the significance of complex roots in polynomial equations?
5. How do you verify if all roots of a polynomial have been found?

**Tip:** When solving cubic polynomials, always check if synthetic division can simplify the process, especially after finding one root.

Find all solutions for the polynomial function f(x) = x^3 + 3x^2 - 2x - 4.

Learn how to solve the cubic polynomial f(x) = x^3 + 3x^2 - 2x - 4. This step-by-step solution includes the application of the Rational Root Theorem, polynomial division, and the quadratic formula. Suitable for students in Grades 9-12.

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