The problem is to simplify the expression $\frac{-4}{-3\sqrt{5}-2}$. This involves rationalizing the denominator.

1. Problem Description

The problem is to simplify the expression $\frac{-4}{-3\sqrt{5}-2}$. This involves rationalizing the denominator.

2. Solution Steps

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The denominator is $-3\sqrt{5} - 2$, so its conjugate is $-3\sqrt{5} + 2$.

Multiply the numerator and denominator by $-3\sqrt{5} + 2$:

$\frac{-4}{-3\sqrt{5}-2} = \frac{-4}{-3\sqrt{5}-2} \cdot \frac{-3\sqrt{5}+2}{-3\sqrt{5}+2}$

$= \frac{-4(-3\sqrt{5}+2)}{(-3\sqrt{5}-2)(-3\sqrt{5}+2)}$

$= \frac{12\sqrt{5}-8}{(-3\sqrt{5})^2 - (2)^2}$

$= \frac{12\sqrt{5}-8}{9(5) - 4}$

$= \frac{12\sqrt{5}-8}{45-4}$

$= \frac{12\sqrt{5}-8}{41}$

3. Final Answer

The simplified expression is $\frac{12\sqrt{5}-8}{41}$.