Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ?
Since $P(0) = k,$ the polynomial $P(x)$ is of the form
\[P(x) = ax^3 + bx^2 + cx + k.\]Since $P(1) = 2k,$
\[a + b + c + k = 2k,\]so $a + b + c = k.$

Since $P(-1) = 3k,$
\[-a + b - c + k = 3k,\]so $-a + b - c = 2k.$   Adding the equations $a + b + c = k$ and $-a + b - c = 2k,$ we get $2b = 3k,$ so $b = \frac{3}{2} k.$

Then
\begin{align*}
P(2) + P(-2) &= (8a + 4b + 2c + k) + (-8a + 4b - 2c + k) \\
&= 8b + 2k = 12k + 2k = \boxed{14k}.
\end{align*}