USAMO 2016 — Days 1 and 2 Question 4

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all real numbers $x$ and $y$, $$ (f(x)+xy) \cdot f(x-3y) + (f(y)+xy) \cdot f(3x-y) = (f(x+y))^2. $$

Proposed by Titu Andreescu (credited by Evan Chen).