jmc

By Factor Theorem, the polynomial will become 0 when $x=-2$ and $x=1.$ Thus, $$\begin{array}{rl}(-2{)}^5-(-2{)}^4+(-2{)}^3-p(-2{)}^2+q(-2)+4& =0,\\ 1-1+1-p+q+4& =0.\end{array}$$Then $-4p-2q=52$ and $-p+q=-5.$ Solving, we find $(p,q)=\boxed{(-7,-12)}.$