For any homogeneous polynomial $f \in \mathbb R [x,y]$, define the homogeneous polynomial

$$H(f) := \partial_yf^2\partial_x\partial_xf-2\partial_xf\partial_yf\;\partial_x\partial_yf+\partial_xf^2\partial_y\partial_yf$$

which is the Hessian of $f$ applied to the tangent of its level set. I came across it while studying the asymptotics of certain complete open surfaces.

From numerical tests, it appears that any homogeneous polynomial $f(x,y)$ factors as follows

$$H(f) = G(x,y) \, f$$

for some homogeneous polynomial $G$.

Question:Is there a simple proof of, or a counterexample to, this factorization?