MathNet 2024-part-03 — 168 problems (EN–VI) Question 29

Find all the functions $f : \mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ we have:
$$
f(yf(x) + f(x)f(y)) = xf(y) + f(xy)
$$