To simplify the expression [tex]\(\frac{x^2 + 5x - 36}{x^2 - 16}\)[/tex], we need to factorize both the numerator and the denominator and then reduce the fraction if possible.
1. Factorize the numerator [tex]\(x^2 + 5x - 36\)[/tex]:
- We need to find two numbers that multiply to [tex]\(-36\)[/tex] and add up to [tex]\(5\)[/tex].
- The numbers [tex]\(9\)[/tex] and [tex]\(-4\)[/tex] fit this requirement: [tex]\(9 \times (-4) = -36\)[/tex] and [tex]\(9 + (-4) = 5\)[/tex].
- Therefore, the numerator can be factorized as:
[tex]\[
x^2 + 5x - 36 = (x + 9)(x - 4)
\][/tex]
2. Factorize the denominator [tex]\(x^2 - 16\)[/tex]:
- Notice that [tex]\(x^2 - 16\)[/tex] is a difference of squares.
- The difference of squares can be factorized as:
[tex]\[
x^2 - 16 = (x + 4)(x - 4)
\][/tex]
3. Rewrite the fraction using the factorizations:
[tex]\[
\frac{x^2 + 5x - 36}{x^2 - 16} = \frac{(x + 9)(x - 4)}{(x + 4)(x - 4)}
\][/tex]
4. Simplify the fraction:
- Notice that [tex]\((x - 4)\)[/tex] is a common factor in both the numerator and the denominator.
- Cancel out the common factor:
[tex]\[
\frac{(x + 9)(x - 4)}{(x + 4)(x - 4)} = \frac{x + 9}{x + 4} \quad \text{for} \quad x \neq 4
\][/tex]
Thus, the simplest form of [tex]\(\frac{x^2 + 5x - 36}{x^2 - 16}\)[/tex] is:
[tex]\[
\boxed{\frac{x+9}{x+4}}
\][/tex]
Therefore, the correct choice is:
[tex]\(\frac{x+9}{x+4}\)[/tex].
