Number of pairs of rational numbers that satisfy the given relation The number of pairs $(x,y)$ that satisfy : $2x^2 + y^2 + 2xy - 2y + 2 = 0$ is
a.) $0$
b.) $1$
c.) $2$
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. 

(in the spirit of the short demonstration of @Takahiro Waki : no need to extract roots...) yet another factorization : $(y + x - 1)^2 + (x + 1)^2$ wich can be null only if the content of the two parentheses are simultaneously equal to zero i.e, for $x=-1$ and $y=2$.
