To solve the equation \( x^4 - 2x^3 - 11x^2 + 12x = 0 \) given \( x = -3 \) as a root, follow these steps:

### Step 1: Factor out the common factor
First, observe that there is a common factor of \( x \) in the equation:
\[
x(x^3 - 2x^2 - 11x + 12) = 0
\]
This gives us one root immediately:
\[
x = 0
\]

### Step 2: Perform synthetic division
Next, use synthetic division to divide \( x^3 - 2x^2 - 11x + 12 \) by \( x + 3 \), since \( x = -3 \) is given as a root.

**Synthetic Division:**

\[
\begin{array}{r|rrrr}
-3 & 1 & -2 & -11 & 12 \\
   &   & -3 & 15  & -12 \\
\hline
   & 1 & -5 & 4 & 0 \\
\end{array}
\]

The quotient from the synthetic division is \( x^2 - 5x + 4 \) with no remainder.

So, \( x^3 - 2x^2 - 11x + 12 = (x + 3)(x^2 - 5x + 4) \).

### Step 3: Factor the quadratic
Now, factor the quadratic equation \( x^2 - 5x + 4 \):
\[
x^2 - 5x + 4 = (x - 4)(x - 1)
\]

### Step 4: Combine all factors
Combining everything, the factored form of the original equation is:
\[
x(x + 3)(x - 4)(x - 1) = 0
\]

### Step 5: Solve for \( x \)
Setting each factor to zero gives the solutions:
\[
x = 0, \, x = -3, \, x = 4, \, x = 1
\]

Thus, the roots of the equation are \( x = 0, -3, 4, 1 \).

---

Would you like more details on any step, or do you have any other questions?

Here are 5 related questions:
1. How do you perform synthetic division in more complex polynomials?
2. What are some methods to factor higher-degree polynomials?
3. How does the Fundamental Theorem of Algebra apply to this problem?
4. Can you explain how to factor polynomials with complex roots?
5. What are the different types of polynomial roots (real, repeated, complex)?

**Tip:** Always check your factored form by expanding it back to ensure it matches the original polynomial.

Learn how to solve the polynomial equation x^4 - 2x^3 - 11x^2 + 12x = 0 step-by-step with synthetic division and factoring. This problem involves polynomials and quadratic equations, suitable for high school students.

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