prove the minimum value by completing square

let $x,y$be positive poof that \[x^3+y^3+xy^2+x^2y-2x^2-2y^2-2xy+2\geqslant0\]

Poof: it can be written as \[\frac{1}{4}(x-1)^2(3x+y)+\frac{1}{4}(y-1)^2(3y+x)+\frac{1}{4}(x+y+2)(x-2+y)^2\], done