1. **State the problem:** We are given the equation $$xy^3 - 4x^2 = 10y^2$$ and we want to analyze or possibly solve it for one variable in terms of the other. 2. **Rewrite the equation:** $$xy^3 - 4x^2 = 10y^2$$ 3. **Attempt to solve for $x$ in terms of $y$: Rewrite terms involving $x$ on one side:** $$xy^3 = 10y^2 + 4x^2$$ 4. This is a nonlinear equation in terms of $x$ and $y$. Try expressing $x$ explicitly by isolating terms. 5. Rearrange to isolate $x$ terms on one side: $$xy^3 - 4x^2 = 10y^2$$ $$x y^3 = 10 y^2 + 4 x^2$$ 6. Let us move all terms to one side: $$0 = 10 y^2 + 4 x^2 - x y^3$$ 7. Treating this as a quadratic in $x$: $$4 x^2 - y^3 x + 10 y^2 = 0$$ 8. Use the quadratic formula for $x$: $$x = \frac{y^3 \pm \sqrt{(y^3)^2 - 4 \cdot 4 \cdot 10 y^2}}{2 \cdot 4}$$ 9. Simplify under the square root: $$x = \frac{y^3 \pm \sqrt{y^6 - 160 y^2}}{8}$$ 10. Factor $y^2$ inside the root: $$x = \frac{y^3 \pm y \sqrt{y^4 - 160}}{8}$$ 11. **Final expression:** $$x = \frac{y^3 \pm y \sqrt{y^4 - 160}}{8}$$ This shows how $x$ is related to $y$ with possible restrictions where the discriminant is non-negative: $$y^4 - 160 \geq 0 \implies y^4 \geq 160$$ So solutions are valid where $|y| \geq \sqrt[4]{160}$. **Summary:** We solved the equation for $x$ in terms of $y$ and found the domain constraints for real solutions.