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**zebra2147** My professor has this proof in his notes and I was wondering if anyone could help get me rolling...

Let $\displaystyle f :\mathbb{R} \rightarrow \mathbb{R}$ be differentiable. Suppose

$\displaystyle f(x + y) = f(x) + f(y) + 2xy$ for all $\displaystyle x,y\in \mathbb{R}$.

Show by completing the two parts below that $\displaystyle f$ is a polynomial of degree 2.

(1) Prove $\displaystyle f'(x) = f'(0)+2x$, for all $\displaystyle x\in \mathbb{R}$.

(2) Prove $\displaystyle f(x) = x^2+f'(0)x+f(0)$,for all $\displaystyle x\in \mathbb{R}$.

I think my biggest problem is coming up with the best way to represent $\displaystyle f'(x)$ which is obviously a big roadblock.