Functions $f$ which satisfy $f(x^4+2x^2+2)-1=(f(x)-1)^4+4x(f(x)-1)^2+4x^2$

Question

I want to find all functions $f:\mathbb R\to \mathbb R$ which satisfy $$f(x^4+2x^2+2)-1=(f(x)-1)^4+4x(f(x)-1)^2+4x^2$$ for all $x$.

I know the solution: the only solution is $f(x)=x$. How can I prove it?