1. **State the problem:** Solve the cubic equation $$6x^3 + 25x^2 - 24x + 5 = 0$$ given that one rational root is 5. 2. **Use the Factor Theorem:** Since 5 is a root, $(x - 5)$ is a factor of the polynomial. 3. **Divide the polynomial by $(x - 5)$:** Use synthetic division or polynomial long division. Synthetic division setup: Coefficients: 6, 25, -24, 5 Bring down 6. Multiply 6 by 5 = 30; add to 25 = 55. Multiply 55 by 5 = 275; add to -24 = 251. Multiply 251 by 5 = 1255; add to 5 = 1260. Since the remainder is not zero, 5 is not a root. This contradicts the given information. 4. **Check the root:** Substitute $x=5$ into the polynomial: $$6(5)^3 + 25(5)^2 - 24(5) + 5 = 6(125) + 25(25) - 120 + 5 = 750 + 625 - 120 + 5 = 1260$$ Since this is not zero, 5 is not a root. 5. **Conclusion:** The given root 5 is incorrect. The solution set cannot be determined from this information. **Final answer:** The root 5 is not a root of the polynomial, so the solution set is empty based on this root.