To simplify the expression [tex]\(\frac{x^2 - x - 6}{x^2 - 9}\)[/tex], let's follow these steps:
1. Factor the numerator and the denominator:
First, let's factor the numerator [tex]\(x^2 - x - 6\)[/tex].
Notice that [tex]\(x^2 - x - 6\)[/tex] can be factored into:
[tex]\[
(x - 3)(x + 2)
\][/tex]
because
[tex]\[
(x - 3)(x + 2) = x^2 + 2x - 3x - 6 = x^2 - x - 6
\][/tex]
Next, let's factor the denominator [tex]\(x^2 - 9\)[/tex].
Notice that [tex]\(x^2 - 9\)[/tex] is a difference of squares, and it factors into:
[tex]\[
(x - 3)(x + 3)
\][/tex]
because
[tex]\[
(x - 3)(x + 3) = x^2 + 3x - 3x - 9 = x^2 - 9
\][/tex]
2. Rewrite the original expression with the factored forms:
[tex]\[
\frac{x^2 - x - 6}{x^2 - 9} = \frac{(x - 3)(x + 2)}{(x - 3)(x + 3)}
\][/tex]
3. Simplify the expression by canceling common factors:
We can cancel the common factor [tex]\((x - 3)\)[/tex] from the numerator and the denominator, as long as [tex]\(x \neq 3\)[/tex]:
[tex]\[
\frac{(x - 3)(x + 2)}{(x - 3)(x + 3)} = \frac{x + 2}{x + 3} \quad \text{for} \quad x \neq 3
\][/tex]
4. State the final simplified expression:
The simplified expression is:
[tex]\[
\frac{x + 2}{x + 3}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{x+2}{x+3}}
\][/tex]
So the correct option from the given choices is C.
