The problem in the image asks to rationalize the denominator and simplify the expression completely. The given expression is:

$\frac{5i}{-2 - 6i}$

To rationalize the denominator, we multiply both the numerator and denominator by the complex conjugate of the denominator, which is $-2 + 6i$. The complex conjugate is used because multiplying a complex number by its conjugate results in a real number.

So, multiply the fraction by $\frac{-2 + 6i}{-2 + 6i}$:

$\frac{5i}{-2 - 6i} \times \frac{-2 + 6i}{-2 + 6i} = \frac{5i(-2 + 6i)}{(-2 - 6i)(-2 + 6i)}$

Now let's expand both the numerator and denominator.

Numerator: $5i(-2 + 6i) = 5i(-2) + 5i(6i) = -10i + 30i^2$ Since $i^2 = -1$, we get: $-10i + 30(-1) = -10i - 30$ So the numerator is $-30 - 10i$.

Denominator: $(-2 - 6i)(-2 + 6i) = (-2)^2 - (6i)^2 = 4 - 36i^2$ Since $i^2 = -1$, we get: $4 - 36(-1) = 4 + 36 = 40$

Thus, the expression becomes: $\frac{-30 - 10i}{40}$

Now simplify the fraction: $\frac{-30}{40} - \frac{10i}{40} = -\frac{3}{4} - \frac{i}{4}$

So the simplified expression is: $-\frac{3}{4} - \frac{i}{4}$

This corresponds to one of the answer choices. Would you like to dive deeper into any steps?