AMC 12 2019a Problem 14

For a certain complex number $c$, the polynomial $$P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)$$has exactly 4 distinct roots. What is $|c|$?

$\textbf{(A) } 2 \textbf{(B) } \sqrt{6} \textbf{(C) } 2\sqrt{2} \textbf{(D) } 3 \textbf{(E) } \sqrt{10}$