2014 AMC 12B Problem 16

Let P be a cubic polynomial with P(0) = k, P(1) = 2k, and P(-1) = 3k. What is P(2) + P(-2) ?

\textbf{(A)}\ 0\qquad\textbf{(B)}\ k\qquad\textbf{(C)}\ 6k\qquad\textbf{(D)}\ 7k\qquad\textbf{(E)}\ 14k

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.