Given that

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

compute $x^2y+xy^2$.
To solve this problem, I will try to eliminate one of the variables and solve for the other. I notice that the first equation can be rewritten as $\frac{x+y}{xy}=5$, which means that $x+y=5xy$. I can substitute this expression into the second equation, and get $3xy+5xy=4$, or $8xy=4$. This implies that $xy=\frac{1}{2}$. Now, I want to find $x^2y+xy^2$, which can be factored as $xy(x+y)$. I already know that $xy=\frac{1}{2}$ and $x+y=5xy=\frac{5}{2}$. Therefore, $x^2y+xy^2=xy(x+y)=\frac{1}{2}\cdot\frac{5}{2}=\frac{5}{4}$.

