problem: Find $p+q$, given that $(q+4)(p+3)-(q+2)(p+1)=44$.
solution: We multiply the binomials:
  
  \begin{align*}
  (q+4)(p+3)-(q+2)(p+1)&=44\\
  q(p+3)+4(p+3)-(q(p+1)+2(p+1))&=44\\
  pq+3q+4p+12-(pq+q+2p+2)&=44\\
  2q+2p+10&=44\\
  2q+2p&=34\\
  q+p&=\boxed{17}\\
  \end{align*}