The problem requires simplifying the expression $\frac{-15xy^5}{10x^4y^4}$ completely.

1. Problem Description

The problem requires simplifying the expression $\frac{-15xy^5}{10x^4y^4}$ completely.

2. Solution Steps

First, we simplify the numerical coefficients:

$\frac{-15}{10} = \frac{-3 \times 5}{2 \times 5} = -\frac{3}{2}$.

Next, we simplify the $x$ terms using the rule $\frac{x^a}{x^b} = x^{a-b}$:

$\frac{x}{x^4} = x^{1-4} = x^{-3} = \frac{1}{x^3}$.

Then, we simplify the $y$ terms using the same rule:

$\frac{y^5}{y^4} = y^{5-4} = y^1 = y$.

Now, we combine the simplified parts:

$-\frac{3}{2} \times \frac{1}{x^3} \times y = -\frac{3y}{2x^3}$.

3. Final Answer

The fully simplified expression is $-\frac{3y}{2x^3}$.