1. **Stating the problem:** Simplify the expression $$\frac{x^4 - x^2}{5x + 5} : \frac{x - 1}{10x}$$. 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{x^4 - x^2}{5x + 5} \times \frac{10x}{x - 1}$$ 3. **Factor where possible:** - Numerator of the first fraction: $$x^4 - x^2 = x^2(x^2 - 1) = x^2(x - 1)(x + 1)$$ - Denominator of the first fraction: $$5x + 5 = 5(x + 1)$$ So the expression becomes: $$\frac{x^2(x - 1)(x + 1)}{5(x + 1)} \times \frac{10x}{x - 1}$$ 4. **Combine the fractions:** $$\frac{x^2(x - 1)(x + 1) \times 10x}{5(x + 1)(x - 1)}$$ 5. **Cancel common factors:** Both numerator and denominator have $(x + 1)$ and $(x - 1)$: $$\frac{x^2 \cancel{(x - 1)} \cancel{(x + 1)} \times 10x}{5 \cancel{(x + 1)} \cancel{(x - 1)}}$$ 6. **Simplify the constants and remaining terms:** $$\frac{x^2 \times 10x}{5} = \frac{10x^3}{5}$$ 7. **Simplify the fraction:** $$\frac{\cancel{10}x^3}{\cancel{5}} = 2x^3$$ **Final answer:** $$2x^3$$