To simplify the given expression [tex]\(\frac{x^2 - x - 6}{x^2 - 9}\)[/tex], follow these steps:
1. Factorize the numerator:
The numerator is [tex]\(x^2 - x - 6\)[/tex]. To factorize it, we need to find two numbers that multiply to [tex]\(-6\)[/tex] and add up to [tex]\(-1\)[/tex].
The numbers [tex]\(-3\)[/tex] and [tex]\(2\)[/tex] satisfy these conditions because [tex]\(-3 \cdot 2 = -6\)[/tex] and [tex]\(-3 + 2 = -1\)[/tex].
So, we can factorize [tex]\(x^2 - x - 6\)[/tex] as:
[tex]\[
x^2 - x - 6 = (x - 3)(x + 2)
\][/tex]
2. Factorize the denominator:
The denominator is [tex]\(x^2 - 9\)[/tex]. This is a difference of squares, which can be factorized as follows:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
3. Rewrite the expression with the factorized numerator and denominator:
[tex]\[
\frac{x^2 - x - 6}{x^2 - 9} = \frac{(x - 3)(x + 2)}{(x - 3)(x + 3)}
\][/tex]
4. Simplify the fraction by canceling out the common factor [tex]\((x - 3)\)[/tex]:
Since [tex]\(x \neq 3\)[/tex] (to avoid division by zero), we can cancel out the common factor [tex]\((x - 3)\)[/tex] from the numerator and denominator:
[tex]\[
\frac{(x - 3)(x + 2)}{(x - 3)(x + 3)} = \frac{x + 2}{x + 3}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{x + 2}{x + 3}
\][/tex]
Hence, the correct option is C.
[tex]\[
\frac{x+2}{x+3}
\][/tex]
