We are asked to simplify the rational expression $\frac{6x^2 - 54}{x^2 + 7x + 12}$.

1. Problem Description

We are asked to simplify the rational expression $\frac{6x^2 - 54}{x^2 + 7x + 12}$.

2. Solution Steps

First, we factor the numerator and the denominator.

The numerator is $6x^2 - 54$. We can factor out a 6:

$6x^2 - 54 = 6(x^2 - 9)$.

Then we can factor the difference of squares $x^2 - 9$:

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

Therefore, the numerator is $6(x - 3)(x + 3)$.

The denominator is $x^2 + 7x + 12$. We look for two numbers that multiply to 12 and add to

7. These numbers are 3 and

4. So we have:

$x^2 + 7x + 12 = (x + 3)(x + 4)$.

Now we can write the expression as

$\frac{6(x - 3)(x + 3)}{(x + 3)(x + 4)}$.

We can cancel the common factor $(x + 3)$:

$\frac{6(x - 3)(x + 3)}{(x + 3)(x + 4)} = \frac{6(x - 3)}{x + 4}$.

3. Final Answer

$\frac{6(x - 3)}{x + 4}$