1. **State the problem:** We are given the implicit equation $$xy^3 - 4x^2 = 10y^2$$ and want to analyze its components. 2. **Rewrite the equation:** The equation relates $x$ and $y$ with polynomial terms: $$xy^3 - 4x^2 = 10y^2$$ 3. **Rearrange to isolate terms:** Move all terms to one side to see the curve more clearly: $$xy^3 - 10y^2 - 4x^2 = 0$$ 4. **Check the degree:** The highest powers are $y^3$ and $x^2$, so it is a polynomial of degree 3 in $y$ and 2 in $x$. 5. **Possible approach to analyze:** It's an implicit polynomial, not easily solved for $y$ explicitly. 6. **Consider intercepts:** - At $x=0$: $$0 - 4(0)^2 = 10y^2 \Rightarrow 0 = 10y^2 \Rightarrow y=0$$ - At $y=0$: $$x(0)^3 - 4x^2 = 10(0)^2 \Rightarrow -4x^2 = 0 \Rightarrow x=0$$ So the curve passes through the origin $(0,0)$. 7. **No easy direct factorization or explicit $y=f(x)$ form:** This suggests implicit plot is necessary for visualization. **Final answer:** The implicit polynomial curve defined by $$xy^3 - 4x^2 = 10y^2$$ passes through origin and is a third degree polynomial in $y$ and second degree in $x$, representing a complex implicit curve.