1. **State the problem:** Find the roots of the equation $$x^3 - 4x^2 + x + 14 = 8$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$x^3 - 4x^2 + x + 14 - 8 = 0$$ which simplifies to $$x^3 - 4x^2 + x + 6 = 0$$. 3. **Use the Rational Root Theorem:** Possible rational roots are factors of the constant term 6 divided by factors of the leading coefficient 1, i.e., $$\pm1, \pm2, \pm3, \pm6$$. 4. **Test possible roots:** - For $$x=1$$: $$1 - 4 + 1 + 6 = 4 \neq 0$$ - For $$x=-1$$: $$-1 - 4 + (-1) + 6 = 0$$, so $$x = -1$$ is a root. 5. **Factor out $$x + 1$$:** Use polynomial division or synthetic division to divide $$x^3 - 4x^2 + x + 6$$ by $$x + 1$$: The quotient is $$x^2 - 5x + 6$$. 6. **Solve the quadratic:** $$x^2 - 5x + 6 = 0$$ Factor: $$(x - 2)(x - 3) = 0$$ So, $$x = 2$$ or $$x = 3$$. 7. **Final roots:** $$x = -1, 2, 3$$. These are the roots of the original equation.