Let's simplify the given rational expression:

$\frac{x^2 - 9}{x^2 - x - 6}$

Step 1: Factor the numerator and denominator.

Factor the numerator:

The numerator is $x^2 - 9$, which is a difference of squares. It can be factored as:

$x^2 - 9 = (x - 3)(x + 3)$

Factor the denominator:

The denominator is $x^2 - x - 6$. We need to find two numbers that multiply to $-6$ and add up to $-1$. These numbers are $-3$ and $2$. Therefore, the denominator factors as:

$x^2 - x - 6 = (x - 3)(x + 2)$

Step 2: Simplify the expression.

Now, substitute the factored forms of the numerator and denominator:

$\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)}$

You can cancel the common factor of $(x - 3)$ from both the numerator and denominator (as long as $x \neq 3$ to avoid division by zero). This leaves:

$\frac{x + 3}{x + 2}$

Step 3: Identify restrictions on $x$.

The original expression has the denominator $x^2 - x - 6$, which factors as $(x - 3)(x + 2)$. To avoid division by zero, $x$ cannot be equal to 3 or -2.

Final Simplified Expression:

$\frac{x + 3}{x + 2}$

Restrictions:

$x \neq 3 \quad \text{and} \quad x \neq -2$

---

Would you like more details on any part of this process or have any questions? Here are some related questions you may find interesting:

1. How do you handle higher-degree polynomials when factoring rational expressions?
2. What happens if we have an expression with multiple factors in both the numerator and denominator?
3. Can we cancel terms when one of the factors is a sum or difference, like in the difference of squares?
4. How can we graph the simplified rational expression and account for its restrictions?
5. How would the result change if the expression had more complex terms in the numerator or denominator?

Tip: Always check for common factors first before simplifying rational expressions, as this helps avoid unnecessary mistakes and ensures you don't miss restrictions on the variable.