Step 1: Find the least common denominator (LCD) for the two fractions. The denominators are $uv$ and $v^2$. The LCD of $uv$ and $v^2$ is $uv^2$. Step 2: Rewrite each fraction with the LCD. For the first fraction, multiply the numerator and denominator by $v$: $$ \frac{u^2 - v^2}{uv} = \frac{(u^2 - v^2) \cdot v}{uv \cdot v} = \frac{u^2v - v^3}{uv^2} $$ For the second fraction, multiply the numerator and denominator by $u$: $$ \frac{v - 3uv - u^2}{v^2} = \frac{(v - 3uv - u^2) \cdot u}{v^2 \cdot u} = \frac{uv - 3u^2v - u^3}{uv^2} $$ Step 3: Add the rewritten fractions. $$ \frac{u^2v - v^3}{uv^2} + \frac{uv - 3u^2v - u^3}{uv^2} = \frac{u^2v - v^3 + uv - 3u^2v - u^3}{uv^2} $$ Step 4: Combine like terms in the numerator. Combine the terms involving $u^2v$: $u^2v - 3u^2v = -2u^2v$. Rearrange the terms in the numerator: $$ -u^3 - 2u^2v + uv - v^3 $$ Step 5: Write the simplified expression. $$ \frac{-u^3 - 2u^2v + uv - v^3}{uv^2} $$ The final answer is $\boxed{\frac{-u^3 - 2u^2v + uv - v^3}{uv^2}}$.