4.1. Given is the polynomial $P(x)=x^{2}-\alpha$, where $\alpha$ is a real number. Determine all polynomials of the form $Q(x)=x^{3}+a x^{2}+b x+c$, where $a, b$, and $c$ are real numbers, such that for every real number $x$, $P(Q(x))=Q(P(x))$ holds.
4.1. $P(Q(x))=x^{6}+2 a x^{5}+\left(a^{2}+2 b\right) x^{4}+(2 c+2 a b) x^{3}+$

$$
+\left(b^{2}+2 a c\right) x^{2}+2 b c x+c^{2}-c
$$

$$
\begin{aligned}
Q(P(x))=x^{6}+(a-3 \alpha) x^{4}+\left(3 \alpha^{2}-2 a \alpha+b\right) x^{2}+ \\
+c-\alpha^{3}-\alpha b .
\end{aligned}
$$

By comparing the coefficients of the odd powers, we get $a=0, c=0$. Now we have the system:

$2 b=-3 \alpha, b^{2}-b=3 \alpha^{2}, \alpha b=\omega-\alpha^{3}$. For $\alpha=0$ we get the solution $b=0$. Otherwise, for $c=2$ the solution is $b=-3$.