2x2 + y2 + 3xy + 3x + 2y + 2 = 0
=> 16x2 + 8y2 + 24xy + 24x + 16y + 16 = 0
=> (4x)2 + 24x(y+1) + 8y2 + 16y + 16 = 0
=> (4x)2 + 24x(y+1) + [3(y + 1)]2 - [3(y + 1)]2 + 8y2 + 16y + 16 = 0
=> (4x + 3y + 3)2 - 9y2 - 18y - 9 + 8y2 + 16y + 16 = 0
=> (4x + 3y + 3)2 - y2 - 2y - 1 + 8 = 0
=> (4x + 3y + 3)2 - (y + 1)2 = -8
=> (y + 1)2 - (4x + 3y + 3)2 = 8
=> (y + 1 +4x + 3y + 3)(y + 1 - 4x - 3y - 3) = 8
=> 4(x + y + 4)( - 4x - 2y - 2) = 8
=> (x + y + 4)( 2x + y + 1) = -1
=> \(\left\{{}\begin{matrix}\left\{{}\begin{matrix}x+y+4=-1\\2x+y+1=1\end{matrix}\right.\\\left\{{}\begin{matrix}x+y+4=1\\2x+y+1=-1\end{matrix}\right.\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}\left\{{}\begin{matrix}x=2\\y=-4\end{matrix}\right.\\\left\{{}\begin{matrix}x=-2\\y=2\end{matrix}\right.\end{matrix}\right.\)
