x^2+2y^2+2x+2=2xy
(x-y)^2+y^2+2x+2=0
(x-y)^2+(y+1)^2+2x-2y+1=0
(x-y)^2+2(x-y)+1+(y+1)^2=0
[(x-y)+1]^2+(y+1)^2=0
(x-y)+1=0
y+1=0
y=-1
x=-2