problem: Given that
  
  \begin{align*}
  \frac{1}{x}+\frac{1}{y}&=5,\\
  3xy+x+y&=4,
  \end{align*}
  
  compute $x^2y+xy^2$.
solution: 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 $\frac{1}{2}\left(\frac{5}{2}\right)=\boxed{\frac{5}{4}}$.