$P(Q(x))$ is also irreducible

Question

For every polynomials $P(x)$ and $Q(x)$ with integer coefficients , if $P(x)$ is irreducible over the rational numbers, then is it true that $P(Q(x))$ is also irreducible over the rational numbers? If $Q(x)$ is also irreducible, will P(Q(x)) be irreducible?

Answer 1

$P(X)=X-1$ is irreducible, $Q(X)=X^2+1$ is irreducible, but $P(Q(X))=X^2$ is not.

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If you think thatusing a linear polynomial is cheating: $P(X)= X^2+4X+5$ and $Q(X)=X^3-2$ are irreducible (e.g., because they have degree $\le 3$ and no rational root), then $P(Q(X))=(X^2+1)(X^4-X^2+1)$.