The Chain Rule & Implicit Differentiation · Superchallenge problem

The equation $z^3 + xz + y = 0$ defines $z$ as a smooth function of $(x,y)$ near the point $(x, y, z) = (-2, 1, 1)$. Find the Hessian determinant $z_{xx}z_{yy} - z_{xy}^2$ at $(x, y) = (-2, 1)$.