This problem involves expanding two products of binomials and simplifying the resulting expression. I can use the distributive property to expand the products: $(q+4)(p+3)=qp+3q+4p+12$ and $(q+2)(p+1)=qp+q+2p+2$. Now I can subtract the second product from the first one and equate it to 44: $qp+3q+4p+12-qp-q-2p-2=44$. Simplifying the expression, I get $2q+2p+10=44$. Subtracting 10 from both sides, I get $2q+2p=34$. Dividing both sides by 2, I get $q+p=17$. This is the answer I was looking for. 
Find $p+q$, given that $(q+4)(p+3)-(q+2)(p+1)=44$.