The number of pairs $(x,y)$ that satisfy : $2x^2 + y^2 + 2xy - 2y + 2 = 0$ is
d.) None of the foregoing numbers
My attempt : I am not well versed in number theory , thus I took the most basic approach that I could see , that is I tried to divide the given equation into squares to and see if i could so something from that however I got stuck at $ (x+y)^2 + x^2 - 2y + 2 $
Also I tried putting x = 0 and realised that there exists no real number y which could form the required pair with x = 0 atleast , similarly i could observe the same thing with y = 0.
Please suggest me a solution as well as a more general approach towards solving these kind of problems
My background is a degree in Electrical Engineering , however I have never taken any specific course in number theory.