1. The problem is to understand and analyze the equation $y^2 + 2xy = C$. 2. This is an implicit equation involving variables $x$ and $y$ and a constant $C$. 3. To analyze it, we can try to express $y$ in terms of $x$ or explore its properties. 4. Rearranging the equation: $$y^2 + 2xy - C = 0$$ 5. Treating this as a quadratic equation in $y$, we use the quadratic formula: $$y = \frac{-2x \pm \sqrt{(2x)^2 - 4 \cdot 1 \cdot (-C)}}{2}$$ 6. Simplify under the square root: $$y = \frac{-2x \pm \sqrt{4x^2 + 4C}}{2}$$ 7. Factor out 4 inside the root: $$y = \frac{-2x \pm 2\sqrt{x^2 + C}}{2}$$ 8. Simplify the fraction: $$y = -x \pm \sqrt{x^2 + C}$$ 9. So the solutions for $y$ are: $$y = -x + \sqrt{x^2 + C} \quad \text{or} \quad y = -x - \sqrt{x^2 + C}$$ 10. This shows the implicit equation can be expressed explicitly as two branches depending on the sign. Final answer: $$y = -x \pm \sqrt{x^2 + C}$$