1. **Stating the problem:** Simplify the algebraic expressions and solve the given rational expressions involving $x$. 2. **Expression 1:** Simplify $\frac{14x^2 - 63x}{7x}$. 3. Use the distributive property and factor where possible: $$14x^2 - 63x = 7x(2x - 9)$$ 4. Substitute back: $$\frac{7x(2x - 9)}{7x}$$ 5. Cancel common factors: $$\frac{\cancel{7x}(2x - 9)}{\cancel{7x}} = 2x - 9$$ 6. **Expression 2:** Simplify $\frac{-10x^2 - 40x^2}{2x^2 - 8x}$. 7. Combine like terms in numerator: $$-10x^2 - 40x^2 = -50x^2$$ 8. Factor numerator and denominator: $$\frac{-50x^2}{2x^2 - 8x} = \frac{-50x^2}{2x(x - 4)}$$ 9. Factor numerator: $$-50x^2 = -50x \cdot x$$ 10. Substitute and cancel common factors: $$\frac{-50x \cdot x}{2x(x - 4)} = \frac{\cancel{-50x} \cdot x}{\cancel{2x}(x - 4)} = \frac{-25x}{x - 4}$$ 11. **Expression 3:** Simplify $x \times 9$. 12. Multiply directly: $$x \times 9 = 9x$$ 13. **Summary:** - $\frac{14x^2 - 63x}{7x} = 2x - 9$ - $\frac{-10x^2 - 40x^2}{2x^2 - 8x} = \frac{-25x}{x - 4}$ - $x \times 9 = 9x$ 14. **Additional notes:** - When simplifying rational expressions, always factor numerator and denominator to cancel common factors. - Be careful with domain restrictions: $x \neq 0$ and $x \neq 4$ to avoid division by zero.