MathNet 2007-part-03 — 185 problems (EN–VI) Question 84

Two real numbers $x$ and $y$ are such that $8 y^{4} + 4 x^{2} y^{2} + 4 x y^{2} + 2 x^{3} + 2 y^{2} + 2 x = x^{2} + 1$. Find all possible values of $x + 2 y^{2}$.