1. **State the problem:** Solve the equation $$5x^3 - 45x = 0$$. 2. **Write down the formula and rules:** To solve polynomial equations, factor the expression and set each factor equal to zero. 3. **Factor the equation:** $$5x^3 - 45x = 5x(x^2 - 9) = 0$$ 4. **Set each factor equal to zero:** $$5x = 0 \quad \Rightarrow \quad x = 0$$ $$x^2 - 9 = 0$$ 5. **Solve the quadratic factor:** $$x^2 - 9 = (x - 3)(x + 3) = 0$$ 6. **Set each factor equal to zero:** $$x - 3 = 0 \Rightarrow x = 3$$ $$x + 3 = 0 \Rightarrow x = -3$$ 7. **Final solutions:** $$x = 0, 3, -3$$