1. Let's start by stating the problem: You want complicated math problems solved with detailed steps. 2. Since no specific problem was provided, here is a challenging algebra problem: Solve for $x$ in the equation $$x^3 - 6x^2 + 11x - 6 = 0.$$ 3. First, try to find rational roots using the Rational Root Theorem candidates: factors of constant term 6 over factors of leading coefficient 1, so possible roots are $\pm1, \pm2, \pm3, \pm6$. 4. Test $x=1$: $$1^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0.$$ So $x=1$ is a root. 5. Divide the cubic by $(x - 1)$: Using synthetic division or polynomial long division, The quotient is $$x^2 - 5x + 6.$$ 6. Factor the quadratic: $$x^2 - 5x + 6 = (x - 2)(x - 3).$$ 7. So the roots are $$x=1, 2, 3.$$ 8. Final answer: The solutions to $$x^3 - 6x^2 + 11x - 6 = 0$$ are $$x=1, x=2,$$ and $$x=3.$$