Find all real pairs $(x,y)$ that satisfy the equation $$5x^2 + 5y^2 + 8xy + 2y - 2x + 2 = 0$$
I thought that it could be solved by factoring everything to construct 2 different equations $(e_1)(e_2) = 0$, however, it seems that the factoring doesn't work too well:
The factored equation should be of form: $(ax+by+c)(lx+my+n)=0$, so it is possible to create a system of equations from that:
$Al = 5$
$bm = 5$
$Am + bl = 8$
$An + cl = -2$
$bn + cm = 2$
$cn = 2$
However, after a few hours of meddling with it, I had found no solutions. That is when I started to wonder that perhaps I am doing something wrongly.
P.S. The problem is supposed to be solvable by a high school student, not a graduate in mathematics.