Question 4
Let $\alpha \neq 0$ be a real number. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(x^2 + y^2) = f(x - y)f(x + y) + \alpha y f(y)$$
holds for all $x, y \in \mathbb{R}$.
$$f(x^2 + y^2) = f(x - y)f(x + y) + \alpha y f(y)$$
holds for all $x, y \in \mathbb{R}$.