How we do solve this problem

Question

Given a polynomial $p(x)$ with $(p(x))^{2}+p(x^{2})=2x^{2} \;\forall x \in {R}$. If $p(1)\neq 1$, then find all posible value $p(10)$.

Answer 1

Hint:

$p(x)$ can be at most first degree ( do you see why?), so it has the form $p(x)=ax+b$ and you want

$(ax+b)^2+ax^2+b=2x^2 \quad \forall x \in \mathbb{R},$ with $a+b \ne 1$

can you do from here ?