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, then factor the polynomial. 3. **Apply Rational Root Theorem:** Possible rational roots are factors of the constant term (-75) divided by factors of the leading coefficient (1), so possible roots are $$\pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75$$. 4. **Test roots:** Substitute $x = -3$: $$(-3)^3 + 3(-3)^2 - 25(-3) - 75 = -27 + 27 + 75 - 75 = 0$$ So, $x = -3$ is a root. 5. **Factor out $(x + 3)$:** Use polynomial division or synthetic division to divide the cubic by $(x + 3)$: $$x^3 + 3x^2 - 25x - 75 = (x + 3)(x^2 - 25)$$ 6. **Solve quadratic factor:** Set $x^2 - 25 = 0$: $$x^2 = 25$$ $$x = \pm 5$$ 7. **Final solutions:** The roots of the equation are $$x = -3, 5, -5$$.