RU-ALLRUSSIAN 2023 Problem 91

Given are two monic quadratics $f(x), g(x)$ such that $f, g, f+g$ have two distinct real roots. Suppose that the difference of the roots of $f$ is equal to the difference of the roots of $g$ . Prove that the difference of the roots of $f+g$ is not bigger than the above common difference.