jmc

Since $p(-7)=p(4)=0,$ the quadratic polynomial $p(x)$ is of the form $$p(x)=c(x+7)(x-4),$$for some constant $c.$ To find $c,$ we set $x=5,$ and use the fact that $p(5)=-36$: $$-36=c(12)(1),$$so $c=-3.$ Hence, $p(x)=-3(x+7)(x-4)=\boxed{-3x^2-9x+84}.$