Let $P(x)$ and $Q(x)$ be distinct, non-constant polynomials such that
\[P(Q(x)) = P(x) Q(x)\]for all $x.$  If $P(1) = P(-1) = 100,$ then find the polynomial $Q(x).$
<|im_start|>system
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<|im_start|>user
Let $P(x)$ and $Q(x)$ be distinct, non-constant polynomials such that
\[P(Q(x)) = P(x) Q(x)\]for all $x.$  If $P(1) = P(-1) = 100,$ then find the polynomial $Q(x).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
x^2
Let $P(x)$ and $Q(x)$ be distinct, non-constant polynomials such that
\[P(Q(x)) = P(x) Q(x)\]for all $x.$  If $P(1) = P(-1) = 100,$ then find the polynomial $Q(x).$
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