AMC 12 2009a Problem 21

Let $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are complex numbers. Suppose that

What is the number of nonreal zeros of $x^{12} + ax^8 + bx^4 + c$?

\textbf{(A)}\ 4 \textbf{(B)}\ 6 \textbf{(C)}\ 8 \textbf{(D)}\ 10 \textbf{(E)}\ 12