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The problem involves finding the limit of a rational expression as \(x\) approaches 2. The expression is:

\[
\lim_{{x \to 2}} \frac{x^3 + 3x^2 - 4x - 12}{x - 2}
\]

### Solution:

To solve this limit, we should first check if direct substitution results in an indeterminate form like \(\frac{0}{0}\). Let's substitute \(x = 2\) into the numerator and denominator:

- **Denominator**: \(x - 2 = 2 - 2 = 0\)
- **Numerator**: 
  \[
  x^3 + 3x^2 - 4x - 12 = 2^3 + 3 \cdot 2^2 - 4 \cdot 2 - 12 = 8 + 12 - 8 - 12 = 0
  \]

Both numerator and denominator are zero, which confirms an indeterminate form. To resolve this, we factorize the numerator and then cancel the common factor.

#### Step 1: Factor the numerator

The numerator is a cubic polynomial: 

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

We suspect it has a factor of \(x - 2\). To confirm, we use synthetic division:

1. Coefficients: \(1, 3, -4, -12\)
2. Dividing by \(x - 2\) means we use \(2\) in synthetic division:

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

The bottom row shows the coefficients of the quotient, which are \(1, 5, 6\), corresponding to \(x^2 + 5x + 6\). The remainder is 0, confirming that \(x - 2\) is a factor.

Thus, the factorization is:

\[
x^3 + 3x^2 - 4x - 12 = (x - 2)(x^2 + 5x + 6)
\]

#### Step 2: Simplify the expression

The limit expression becomes:

\[
\lim_{{x \to 2}} \frac{(x - 2)(x^2 + 5x + 6)}{x - 2}
\]

We can cancel the common factor \(x - 2\):

\[
\lim_{{x \to 2}} (x^2 + 5x + 6)
\]

#### Step 3: Evaluate the limit

Now, substitute \(x = 2\) into the simplified expression:

\[
2^2 + 5 \cdot 2 + 6 = 4 + 10 + 6 = 20
\]

### Conclusion:

\[
\lim_{{x \to 2}} \frac{x^3 + 3x^2 - 4x - 12}{x - 2} = 20
\]

Would you like more details or have any questions? 

### Related Questions:

1. How do you perform synthetic division for polynomial factorization?
2. What are some common indeterminate forms encountered in calculus?
3. How can you apply L'Hopital's Rule to limits involving indeterminate forms?
4. How can polynomial long division help in simplifying rational functions?
5. What are the methods to factorize cubic polynomials?

### Tip:

Always check for common factors when simplifying limits of rational functions to avoid indeterminate forms like \(\frac{0}{0}\).

Find the limit of the expression (x^3 + 3x^2 - 4x - 12) / (x - 2) as x approaches 2.

Learn how to calculate the limit of (x^3 + 3x^2 - 4x - 12) / (x - 2) as x approaches 2 using polynomial factorization and synthetic division. This problem covers key calculus concepts and theorems, suitable for students in Grades 10-12.

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