To solve the equation \( x^3 + 3x^2 - 4x = 12 \), we first rewrite it in standard form:

\[ x^3 + 3x^2 - 4x - 12 = 0 \]

This is a cubic equation, and we can start solving it by checking for possible rational roots using the Rational Root Theorem, as before.

### Rational Root Theorem
The possible rational roots are the factors of the constant term \(-12\) divided by the factors of the leading coefficient \(1\). Thus, the possible rational roots are:

\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \]

### Testing Possible Rational Roots

Let's test these possible roots by substituting them into the polynomial equation \( x^3 + 3x^2 - 4x - 12 \):

1. \( x = 1 \):
   \[
   1^3 + 3(1)^2 - 4(1) - 12 = 1 + 3 - 4 - 12 = -12 \neq 0
   \]
   So, \( x = 1 \) is not a root.

2. \( x = -1 \):
   \[
   (-1)^3 + 3(-1)^2 - 4(-1) - 12 = -1 + 3 + 4 - 12 = -6 \neq 0
   \]
   So, \( x = -1 \) is not a root.

3. \( x = 2 \):
   \[
   2^3 + 3(2)^2 - 4(2) - 12 = 8 + 12 - 8 - 12 = 0
   \]
   So, \( x = 2 \) is a root.

Since \( x = 2 \) is a root, we can factor \( x - 2 \) out of the polynomial.

### Factoring \( x^3 + 3x^2 - 4x - 12 \)

Divide the polynomial \( x^3 + 3x^2 - 4x - 12 \) by \( x - 2 \) using synthetic division.

#### Synthetic Division
\[
\begin{array}{r|rrrr}
2 & 1 & 3 & -4 & -12 \\
  &   & 2 & 10 & 12 \\
\hline
  & 1 & 5 & 6 & 0 \\
\end{array}
\]

The quotient is \( x^2 + 5x + 6 \).

So, \( x^3 + 3x^2 - 4x - 12 = (x - 2)(x^2 + 5x + 6) \).

### Solving \( x^2 + 5x + 6 = 0 \)
Factor \( x^2 + 5x + 6 \):
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]

### Complete Factorization
\[
x^3 + 3x^2 - 4x - 12 = (x - 2)(x + 2)(x + 3)
\]

### Finding the Roots
Setting each factor to zero:
\[
x - 2 = 0 \implies x = 2
\]
\[
x + 2 = 0 \implies x = -2
\]
\[
x + 3 = 0 \implies x = -3
\]

Thus, the roots of the equation \( x^3 + 3x^2 - 4x = 12 \) are:
\[
x = 2, -2, -3
\]

Would you like any more details or have any questions?

### Related Questions
1. How can we verify the roots of a polynomial equation?
2. What is the Rational Root Theorem and how is it used?
3. Can you explain synthetic division in detail?
4. What are the other methods to solve cubic equations?
5. How do you factor quadratic polynomials?
6. What are the applications of finding roots of polynomials?
7. How do complex roots of polynomials affect their graphs?
8. Can a cubic equation have more than three real roots?

### Tip
When solving polynomial equations, always check for simple rational roots first to simplify the factorization process.

Learn how to solve the cubic equation x^3 + 3x^2 - 4x = 12 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomial equations, the Rational Root Theorem, and factorization techniques. Suitable for students in Grades 10-12.

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