The problem asks to simplify the rational expression $\frac{2x^2 + 15x + 18}{4x^2 - 9}$.

1. Problem Description

The problem asks to simplify the rational expression $\frac{2x^2 + 15x + 18}{4x^2 - 9}$.

2. Solution Steps

First, we factor the numerator:

$2x^2 + 15x + 18$. We look for two numbers that multiply to $2 * 18 = 36$ and add to $15$. These numbers are $3$ and $12$. Thus we rewrite the middle term as $3x + 12x$:

$2x^2 + 3x + 12x + 18$.

Now we factor by grouping:

$x(2x + 3) + 6(2x + 3) = (2x + 3)(x + 6)$.

Next, we factor the denominator:

$4x^2 - 9$. This is a difference of squares:

$4x^2 - 9 = (2x)^2 - (3)^2 = (2x - 3)(2x + 3)$.

Therefore, the expression becomes:

$\frac{(2x + 3)(x + 6)}{(2x - 3)(2x + 3)}$.

We can cancel the common factor $(2x + 3)$, provided that $2x + 3 \neq 0$, which means $x \neq -\frac{3}{2}$.

$\frac{(2x + 3)(x + 6)}{(2x - 3)(2x + 3)} = \frac{x + 6}{2x - 3}$.

3. Final Answer

$\frac{x + 6}{2x - 3}$