Hessian-Matrix of a Determinant

Question

Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a polynomial function and let $H_f(x)$ denote its Hessian. Now define $p:=\det(H_f(x))$. Is there a nice way to relate the Hessian of $p$ i.e. $H_p(x)$ with the Hessian of $f$?