MathNet 2026 — Part 04 Question 67

Find all $f: \mathbb{R} \longrightarrow \mathbb{R}$ applications such as for all $x, y$ in $\mathbb{R}$, $$
f(x+y)^2 - f\left(2x^2\right) = f(y+x) f(y-x) + 2x f(y)
$$