1. **State the problem:** Simplify or factor the cubic polynomial $2x^3 + 9x^2 - 6x - 5$. 2. **Group terms:** Group the polynomial into two pairs: $$ (2x^3 + 9x^2) + (-6x - 5) $$ 3. **Factor each group:** From the first group, factor out $x^2$: $$ x^2(2x + 9) $$ From the second group, factor out $-1$: $$ -1(6x + 5) $$ 4. **Rewrite the expression:** $$ x^2(2x + 9) - 1(6x + 5) $$ 5. **Check for common binomial factor:** The binomials $2x + 9$ and $6x + 5$ are not the same, so try another approach. 6. **Try rational root theorem:** Possible rational roots are factors of constant term over factors of leading coefficient: $\pm1, \pm\frac{1}{2}, \pm5, \pm\frac{5}{2}$. 7. **Test $x=\frac{1}{2}$:** $$ 2(\frac{1}{2})^3 + 9(\frac{1}{2})^2 - 6(\frac{1}{2}) - 5 = 2(\frac{1}{8}) + 9(\frac{1}{4}) - 3 - 5 = \frac{1}{4} + \frac{9}{4} - 8 = \frac{10}{4} - 8 = 2.5 - 8 = -5.5 \neq 0 $$ 8. **Test $x=-1$:** $$ 2(-1)^3 + 9(-1)^2 - 6(-1) - 5 = -2 + 9 + 6 - 5 = 8 \neq 0 $$ 9. **Test $x=1$:** $$ 2(1)^3 + 9(1)^2 - 6(1) - 5 = 2 + 9 - 6 - 5 = 0 $$ So, $x=1$ is a root. 10. **Divide polynomial by $(x-1)$:** Use synthetic division or polynomial division: $$ \frac{2x^3 + 9x^2 - 6x - 5}{x - 1} = 2x^2 + 11x + 5 $$ 11. **Factor quadratic $2x^2 + 11x + 5$:** Find two numbers that multiply to $2 \times 5 = 10$ and add to $11$: $10$ and $1$. Rewrite: $$ 2x^2 + 10x + x + 5 = 2x(x + 5) + 1(x + 5) = (2x + 1)(x + 5) $$ 12. **Final factorization:** $$ (x - 1)(2x + 1)(x + 5) $$ **Answer:** The factorization of $2x^3 + 9x^2 - 6x - 5$ is $$ (x - 1)(2x + 1)(x + 5) $$