Simplify the expression $\frac{x^2 - 9}{x^2 + 5x + 6}$.

1. Problem Description

Simplify the expression $\frac{x^2 - 9}{x^2 + 5x + 6}$.

2. Solution Steps

First, factor the numerator and the denominator.

The numerator is a difference of squares: $x^2 - 9 = x^2 - 3^2$.

The difference of squares formula is:

$a^2 - b^2 = (a-b)(a+b)$

So, $x^2 - 9 = (x-3)(x+3)$.

The denominator is a quadratic expression: $x^2 + 5x + 6$.

We need to find two numbers that multiply to 6 and add to

5. Those numbers are 2 and

3. So, $x^2 + 5x + 6 = (x+2)(x+3)$.

Now, rewrite the expression with the factored numerator and denominator:

$\frac{x^2 - 9}{x^2 + 5x + 6} = \frac{(x-3)(x+3)}{(x+2)(x+3)}$

Cancel out the common factor $(x+3)$ from the numerator and the denominator:

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

3. Final Answer

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