To solve this question, we aim to simplify the given rational expression and match it to one of the options.

The given expression is:

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

Step 1: Factorize the numerator and denominator

• Numerator: $x^2 - x - 6$ factors as $(x - 3)(x + 2)$.
• Denominator: $x^2 - 9$ is a difference of squares, which factors as $(x - 3)(x + 3)$.

The expression becomes:

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

Step 2: Simplify the expression

For $x > 3$, $x - 3 \neq 0$, so we can cancel $x - 3$ in the numerator and denominator:

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

Step 3: Match the result with the options

The simplified expression is $\frac{x + 2}{x + 3}$, which corresponds to Option E.

Final Answer:

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

Would you like a detailed explanation of the factoring process or additional examples? Here are some related questions to explore:

1. How do you factor quadratic polynomials like $x^2 - x - 6$?
2. What is the significance of the difference of squares in factoring $x^2 - 9$?
3. Why is it valid to cancel out $(x - 3)$ when $x > 3$?
4. How can similar rational expressions be simplified when there are no common terms?
5. Can this type of problem appear in real-world applications of rational expressions?

Tip: Always check for restrictions in the domain when simplifying rational expressions to avoid invalid cancellations.