VWO 1987 Problem 3

Find all continuous functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x)^3 = -\frac x{12}\cdot\left(x^2+7x\cdot f(x)+16\cdot f(x)^2\right),\ \forall x \in \mathbb{R}.$$