<s>[INST] Find all polynomials with real coefficients that satisfy $(x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ Find all polynomials with real coefficients that satisfy $$(x^2-6x+8)P(x)=(x^2+2x)P(x-2)\forall x\in\Bbb R$$
My work;
$$\frac{P(x)}{P(x-2)}=-\frac{4}{x-2}+\frac{12}{x-4}+1\tag{1}$$
$$\frac{P(x-2)}{P(x)}=-\frac{12}{x+2}+\frac{4}{x}+1\tag{2}$$
I also factorised the two known polynomial which didn't give anything useful. What should I extract from these two ratios?
 [/INST] The relation is $(x-2)(x-4)P(x)=x(x+2)P(x-2)$ hence $$\frac{P(x)}{(x+2)x^2(x-2)}=\frac{P(x-2)}{x(x-2)^2(x-4)}.$$ Can you conclude?
</s>