jmc

Note that $p(x)=2x$ for $x=-3,$ 4, and 5, so we consider the polynomial $$q(x)=p(x)-2x,$$which is cubic.

Then $q(-3)=q(4)=q(5)=0,$ so $q(x)$ is of the form $$q(x)=c(x+3)(x-4)(x-5)$$for some constant $c$. Furthermore, $q(7)=15-2\cdot 7=1,$ and $$q(7)=c(7+3)(7-4)(7-5)=60c,$$so $c=\frac{1}{60}.$ Hence, $$q(x)=\frac{(x+3)(x-4)(x-5)}{60}.$$In particular, $$q(12)=\frac{(12+3)(12-4)(12-5)}{60}=14,$$so $p(12)=q(12)+2\cdot 12=\boxed{38}.$