1. **State the problem:** Simplify the expression $$\left(\frac{x^2 - 4x - 5}{x^2 - 3x - 10}\right)\left(\frac{9x + 3x^2}{x^2 - 9}\right)$$ and state the restrictions on $x$. 2. **Factor all polynomials:** - Numerator of first fraction: $$x^2 - 4x - 5 = (x - 5)(x + 1)$$ - Denominator of first fraction: $$x^2 - 3x - 10 = (x - 5)(x + 2)$$ - Numerator of second fraction: $$9x + 3x^2 = 3x(3 + x) = 3x(x + 3)$$ - Denominator of second fraction: $$x^2 - 9 = (x - 3)(x + 3)$$ 3. **Rewrite the expression with factored forms:** $$\left(\frac{(x - 5)(x + 1)}{(x - 5)(x + 2)}\right)\left(\frac{3x(x + 3)}{(x - 3)(x + 3)}\right)$$ 4. **Cancel common factors:** - Cancel $(x - 5)$ from numerator and denominator: $$\frac{\cancel{(x - 5)}(x + 1)}{\cancel{(x - 5)}(x + 2)} = \frac{x + 1}{x + 2}$$ - Cancel $(x + 3)$ from numerator and denominator: $$\frac{3x\cancel{(x + 3)}}{(x - 3)\cancel{(x + 3)}} = \frac{3x}{x - 3}$$ 5. **Multiply the simplified fractions:** $$\frac{x + 1}{x + 2} \times \frac{3x}{x - 3} = \frac{3x(x + 1)}{(x + 2)(x - 3)}$$ 6. **State restrictions:** - Denominators cannot be zero: - From $x^2 - 3x - 10 = (x - 5)(x + 2)$, $x \neq 5$, $x \neq -2$ - From $x^2 - 9 = (x - 3)(x + 3)$, $x \neq 3$, $x \neq -3$ **Final answer:** $$\boxed{\frac{3x(x + 1)}{(x + 2)(x - 3)}}$$ **Restrictions:** $$x \neq 5, -2, 3, -3$$