jmc

Let $q(x)=p(x)-4.$ Then $q(x)$ is a cubic polynomial, and $q(-3)=q(-2)=q(4)=0,$ so $$q(x)=c(x+3)(x+2)(x-4)$$for some constant $c.$ Also, $q(5)=16-4=12,$ and $$q(5)=c(8)(7)(1),$$so $c=\frac{3}{14}.$ Hence, $$q(x)=\frac{3(x+3)(x+2)(x-4)}{14}.$$In particular, $$q(11)=\frac{3\cdot 14\cdot 13\cdot 7}{14}=273,$$so $p(11)=273+4=\boxed{277}.$