<s>[INST] Find all polynomials $Q(x)(x^2-6x+8) =Q(x-2)(x^2-6x)$ for $x\in \Bbb R$ 
Find all polynomials $Q(x)(x^2-6x+8) =Q(x-2)(x^2-6x)$ for $x\in \Bbb R$

I've tried different substitutions like put $Q(x)$ $=$ $k$ for some $k$ and getting the equation $k(x^2-6x+8)$ $=$ $(k-2)(x^2-6x)$ $<=>$ $k$=$(6x-x^2)/4$, but that doesn't give equlity. Thank you for your help
 [/INST] $$\begin{align*}
(x^2-6x+8) Q(x) &\equiv (x^2 -6x)Q(x-2)\\
(x^2 - 6x)[Q(x)-Q(x-2)] &\equiv -8Q(x)
\end{align*}$$
If $Q(x)$ has degree $n > 0$, $Q(x) - Q(x-2)$ has degree $n-1$, and so the left hand side has degree $n+1$. However, the right hand side has degree $n$.
So $n\not>0$, and $Q(x)$ can only be a constant function with degree $0$. Let $Q(x) \equiv q$,
$$\begin{align*}
(x^2-6x+8)q &\equiv (x^2 -6x)q\\
8q &\equiv 0\\
q &\equiv 0\\
Q(x) &\equiv 0
\end{align*}$$

Notes regarding that $Q(x) - Q(x-2)$ has degree $n-1$: Let
$$Q(x) \equiv ax^n + bx^{n-1} + \cdots$$
where $a \ne 0$ and all lower degree terms are omitted. Then $Q(x-2)$ is
$$\begin{align*}
Q(x-2) &\equiv a(x^n - 2nx^{n-1} + \cdots) + b(x^{n-1} - \cdots) + \cdots\\
&\equiv ax^n -2anx^{n-1} + bx^{n-1} + \cdots\\
Q(x)-Q(x-2)&\equiv 2anx^{n-1} + \cdots
\end{align*}$$
Since $a\ne 0$ and $n> 0$, the coefficient of $x^{n-1}$ is non-zero, and so the right hand side has degree $n-1$.
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