1. **State the problem:** Solve the equation $x + 2xy - y^2 = 2$ for $x$ in terms of $y$. 2. **Rewrite the equation:** The equation is $x + 2xy - y^2 = 2$. 3. **Group terms with $x$:** We can factor $x$ from the terms involving it: $$x + 2xy = x(1 + 2y)$$ So the equation becomes: $$x(1 + 2y) - y^2 = 2$$ 4. **Isolate $x(1 + 2y)$:** $$x(1 + 2y) = 2 + y^2$$ 5. **Solve for $x$:** $$x = \frac{2 + y^2}{1 + 2y}$$ 6. **Simplify if possible:** Check if numerator and denominator share common factors. They do not, so this is the simplified form. 7. **Important note:** The denominator $1 + 2y$ cannot be zero, so $y \neq -\frac{1}{2}$. **Final answer:** $$x = \frac{2 + y^2}{1 + 2y}$$