jmc

We multiply the binomials:

$$\begin{array}{rl}(q+4)(p+3)-(q+2)(p+1)& =44\\ q(p+3)+4(p+3)-(q(p+1)+2(p+1))& =44\\ pq+3q+4p+12-(pq+q+2p+2)& =44\\ 2q+2p+10& =44\\ 2q+2p& =34\\ q+p& =\boxed{17}\end{array}$$