1. **State the problem:** Simplify the rational function $$f(x) = \frac{x^3 - x^2 - 12x}{4x^3 - 4x^2 - 8x}$$. 2. **Factor numerator and denominator:** - Numerator: $$x^3 - x^2 - 12x = x(x^2 - x - 12)$$ - Factor quadratic: $$x^2 - x - 12 = (x - 4)(x + 3)$$ - So numerator becomes $$x(x - 4)(x + 3)$$ - Denominator: $$4x^3 - 4x^2 - 8x = 4x(x^2 - x - 2)$$ - Factor quadratic: $$x^2 - x - 2 = (x - 2)(x + 1)$$ - So denominator becomes $$4x(x - 2)(x + 1)$$ 3. **Rewrite the function:** $$f(x) = \frac{x(x - 4)(x + 3)}{4x(x - 2)(x + 1)}$$ 4. **Cancel common factors:** - Both numerator and denominator have a factor of $$x$$, so cancel it: $$f(x) = \frac{\cancel{x}(x - 4)(x + 3)}{4\cancel{x}(x - 2)(x + 1)}$$ 5. **Final simplified form:** $$f(x) = \frac{(x - 4)(x + 3)}{4(x - 2)(x + 1)}$$ This is the simplified form of the given rational function. **Note:** The domain excludes values that make the original denominator zero: $$x \neq 0, 2, -1$$.