Given that

\begin{align*}
\frac{1}{x}+\frac{1}{y}&=5,\\
3xy+x+y&=4,
\end{align*}

compute $x^2y+xy^2$.
The first equation becomes

$$\frac{x+y}{xy}=5\Rightarrow x+y=5xy.$$

Substituting into the second equation,

$$8xy=4\Rightarrow xy=\frac{1}{2}.$$

Thus $x+y=\frac{5}{2}$.

The quantity we desire factors as $xy(x+y)$, so it is equal to $\frac{1}{2}\left(\frac{5}{2}\right)=\boxed{\frac{5}{4}}$.