nobody has any solution? it's strange: 4 years and no solution yet

So, as I understand, $g$ is a polynomial of second degree. Let us consider the case when the senior coefficient of $g$ is positive. The other case is similar. Suppose there exists a point $x_0\in \mathbb{R}$ with $g(x_0)>f(x_0)$. Then consider a line $mx+n$ through $(x_0, f(x_0))$ that does not meet the graph of $g(x)$ (it's possible) . It leads to contradiction. Hence, it holds $f(x)\geq g(x)$ for any $x\in \mathbb{R}$. Suppose there exists $x_0\in \mathbb{R}$ with $f(x_0)>g(x_0)$. Then consider a line $mx+n$ through $(x_0,g(x_0))$ and tangent to the graph of $g(x)$. It again leads to a contradiction. Thus, $f(x)=g(x)\,,\,\forall x\in\mathbb{R}$.