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. The original answer is in \(\sqrt{m}\) format, please give the value of \(m\).

Let's think step by step and output the final answer within \boxed{}.
user