Find a function $f :\mathbb{R} \to \mathbb{R}$ with some conditions

Question

Find a function $f :\mathbb{R} \to \mathbb{R}$ satisfying that : $$f(1)=1$$ $$f(x+y)=f(x)+f(y)+2xy$$ $$f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4} \hspace{5pt}\forall x \neq 0$$

Answer 1

Hint: $$(x+y)^2=x^2+y^2+2xy{}$$

Answer 2

I do to here : let $g(x)=f(x)-x^2$ , at that time : $$g(x+y)=g(x)+g(y)$$ $$g(1)=g(0)=0$$ $$g(\frac{1}{x})=\frac{g(x)}{x^{4}}$$

I proved $g(x)\equiv 0$ with $ x \in \mathbb{Q}$ but proving $g(x) \equiv 0$ with $x \in \mathbb{R}$ is problem for me, can you help me more