1. **State the problem:** Solve the equation $x + 2xy - y^2 = 2$ for $x$ in terms of $y$. 2. **Rewrite the equation:** $$x + 2xy - y^2 = 2$$ 3. **Group terms with $x$:** $$x + 2xy = 2 + y^2$$ 4. **Factor out $x$ on the left side:** $$x(1 + 2y) = 2 + y^2$$ 5. **Divide both sides by $(1 + 2y)$ to isolate $x$:** $$x = \frac{2 + y^2}{1 + 2y}$$ 6. **Show cancellation if possible:** Since there are no common factors to cancel, the expression remains as is. 7. **Final answer:** $$\boxed{x = \frac{2 + y^2}{1 + 2y}}$$ This expresses $x$ explicitly in terms of $y$.