1. **State the problem:** Solve the cubic equation $$x^3 + 3x^2 - 25x - 75 = 0$$. 2. **Formula and approach:** To solve a cubic equation, we can try to find rational roots using the Rational Root Theorem, which suggests possible roots are factors of the constant term divided by factors of the leading coefficient. Here, possible roots are factors of 75 (\pm1, \pm3, \pm5, \pm15, \pm25, \pm75). 3. **Test possible roots:** Substitute these values into the equation to find a root. - Test $x=5$: $$5^3 + 3(5)^2 - 25(5) - 75 = 125 + 75 - 125 - 75 = 0$$ So, $x=5$ is a root. 4. **Factor the cubic:** Use polynomial division or synthetic division to divide the cubic by $(x-5)$: Dividing, we get: $$x^3 + 3x^2 - 25x - 75 = (x - 5)(x^2 + 8x + 15)$$ 5. **Solve the quadratic factor:** $$x^2 + 8x + 15 = 0$$ Use the quadratic formula: $$x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1} = \frac{-8 \pm \sqrt{64 - 60}}{2} = \frac{-8 \pm 2}{2}$$ 6. **Find roots from quadratic:** - $$x = \frac{-8 + 2}{2} = \frac{-6}{2} = -3$$ - $$x = \frac{-8 - 2}{2} = \frac{-10}{2} = -5$$ 7. **Final answer:** The solutions to the equation are: $$x = 5, -3, -5$$