Solution
= Solution
Taking the second <partial derivatives> of the <polynomial> gives
$$
w_{xx}=\frac{180x^2y^2-30x^4+30y^4}{360},\qquad
w_{yy}=\frac{180x^2y^2+30x^4-30y^4}{360}.
$$
Adding cancels the pure fourth powers and leaves
$$
\boxed{w_{xx}+w_{yy}=x^2y^2}.
$$
Thus direct substitution verifies the stated <Poisson equation>.