(x-2)2=x2-4x+4; (y-3)2=y2-6y+9; (x-2)(y-3)=xy-3x-2y+6.
Let x2+xy+y2=A(y2-6y+9)+B(x2-4x+4)+C(xy-3x-2y+6)+D(x-2)+E(y-3)+F, then we need to match the coefficients of x2, x, y2, y, xy and constant.
x2: B=1
y2: A=1
x: -4B-3C+D=0
y: -6A-2C+E=0
xy: C=1
constant: 9A+4B+6C-2D-3E+F=0.
Since we know A=B=C=1, substitute in the other equations:
x: -4-3+D=0, D=7
y: -6-2+E=0, E=8
constant: 9+4+6-14-24+F=0, F=19.
f(x,y)=(y2-6y+9)+(x2-4x+4)+(xy-3x-2y+6)+7(x-2)+8(y-3)+19=
x2+y2+(-4-3+7)x+(-6-2+8)y+xy+(9+4+6-14-24+19)=x2+y2+xy.
So f(x-2,y-3)=(x-2)2+(y-3)2+(x-2)(y-3)+7(x-2)+8(y-3)+19.
f(x,y)=x2+y2+xy+7x+8y+19.