To simplify the rational expression:

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

we need to factor both the numerator and the denominator.

Step 1: Factor the numerator $x^2 - 9$

Notice that $x^2 - 9$ is a difference of squares. It factors as:

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

Step 2: Factor the denominator $x^2 - x - 6$

To factor $x^2 - x - 6$, we look for two numbers that multiply to $-6$ and add to $-1$. These numbers are $-3$ and $2$, so we can factor the quadratic as:

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

Step 3: Simplify the rational expression

Now we can rewrite the expression:

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

We can cancel out the common factor of $x - 3$ from both the numerator and the denominator:

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

Final Answer:

The simplified expression is:

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

Do you have any questions, or would you like further details?

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Related Questions:

1. How do you factor a difference of squares?
2. What are the steps to factor a quadratic expression like $x^2 - x - 6$?
3. What happens when you cancel a factor in a rational expression?
4. How do you handle rational expressions with higher-degree polynomials?
5. How can you check if your factored form is correct?

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Tip:

When factoring quadratics, always check if the coefficient of $x^2$ is 1. If it's not, you may need to use the "ac method" or complete the square!