If $f(x)=x^2+x+2$ and $g(x)=x^2-x+2$, how to prove that there is no function $ h:\mathbb R \to \mathbb R $ such that $h(f)+h(g) = g(f)$?

Question

I tried to substitute $f(x)$ & $g(x)$ in their places but didn't find a relation; The function beginning bijective or surjective etc have nothing to do with our case I believe

$g(f(x))=x^4+2x^3+4x^2+3x+4$

Here is a graph

Answer 1

Suppose that such a function existed.
Then, we would have, for all $x \in \Bbb R$, the identity: $$h(f(x)) + h(g(x)) = g(f(x)).$$

Letting $x = 1$, we get $$h(4) + h(2) = 14.$$

Letting $x = -1$, we get $$h(2) + h(4) = 4.$$

The above two equations clearly lead to a contradiction.