$P(x)$ is a polynomial with real coefficients satisfying $$P(x)P(x+1)=P(x^2+2)$$
I did find some solutions for $P(x)P(x+1)=P(x^2)$ and I wonder if I can do the same for this problem
I tried making $P(x)=a_nx^n+\cdots+a_1x+a_0$, and got that $a_n=1$, but I don't know how to make progress
P/s: I do found that $(x^2-x+2)^n$ is a solution. The solution is here: https://artofproblemsolving.com/community/c6h392562p2239953.