i came across the formula in picrel while reading a proof. not surprisingly, the proof of this formula was left to the reader

for context, we are working with homogeneous polynomials in 3 variables (x,y,z). h represents the Hessian determinant of the function, but in that particular case it's meant to be the dehomogenization of said Hessian determinant (so basically just calculating the determinant as usual and then taking z=1, as far as i understand). my first guess was using Euler's formula for homogeneous polynomials (and its partial derivatives), in order to write all derivatives with respect to z in terms of x and y only. and while it seems like it could work, the determinant becomes an absolutely huge mess that seems like it'd take forever to solve. it doesn't feel like the right approach

has anyone seen that formula before, or have an idea of how to derive it more neatly?