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} $.