To simplify the given rational expression, let us carefully factorize the numerator and the denominator.

Problem:

Simplify: $\frac{x^2 - x - 12}{x^2 + 2x - 24}$

Step-by-Step Solution:

1. Factorize the numerator $(x^2 - x - 12)$: $x^2 - x - 12 = (x - 4)(x + 3)$

2. Factorize the denominator $(x^2 + 2x - 24)$: $x^2 + 2x - 24 = (x + 4)(x - 6)$

3. Rewrite the expression: Substitute the factored forms: $\frac{x^2 - x - 12}{x^2 + 2x - 24} = \frac{(x - 4)(x + 3)}{(x + 4)(x - 6)}$

4. Simplify: No terms cancel between the numerator and denominator, so the simplified form remains: $\frac{(x - 4)(x + 3)}{(x + 4)(x - 6)}$

Final Answer:

The correct option is: C: $\frac{(x - 4)(x + 3)}{(x + 4)(x - 6)}$.

Would you like further clarification or details? Here are some related questions for practice:

1. How do you factor quadratic equations in general?
2. What happens when a term cancels between numerator and denominator?
3. How can you determine if a rational expression is undefined?
4. Why can't terms be canceled if they're not common factors?
5. How would you simplify $\frac{x^2 + 5x + 6}{x^2 + x - 6}$?

Tip: Always check for common factors in both numerator and denominator before simplifying any rational expression.