1. **State the problem:** Simplify the rational expression $$\frac{x^3 - 6x^2 + 11x - 6}{x^3 - 4x^2 + x + 6}$$ and verify it equals $$\frac{(x - 1)(x - 2)(x - 3)}{(x + 1)(x - 2)(x - 3)}$$ and then simplify further to $$\frac{x - 1}{x + 1}$$. 2. **Factor numerator and denominator:** - Numerator: $$x^3 - 6x^2 + 11x - 6$$ factors as $$(x - 1)(x - 2)(x - 3)$$. - Denominator: $$x^3 - 4x^2 + x + 6$$ factors as $$(x + 1)(x - 2)(x - 3)$$. 3. **Write the fraction with factored forms:** $$\frac{(x - 1)(x - 2)(x - 3)}{(x + 1)(x - 2)(x - 3)}$$ 4. **Cancel common factors:** The terms $$(x - 2)$$ and $$(x - 3)$$ appear in both numerator and denominator, so we cancel them: $$\frac{\cancel{(x - 1)}\cancel{(x - 2)}\cancel{(x - 3)}}{(x + 1)\cancel{(x - 2)}\cancel{(x - 3)}} = \frac{x - 1}{x + 1}$$ 5. **Final simplified expression:** $$\boxed{\frac{x - 1}{x + 1}}$$ This shows the original complex rational expression simplifies neatly to $$\frac{x - 1}{x + 1}$$ after factoring and canceling common factors.