1. **State the problem:** Find the root(s) of the polynomial equation $$x(x-2)(x+3) = 18$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x(x-2)(x+3) - 18 = 0$$ 3. **Expand the polynomial:** First, expand $$x(x-2)(x+3)$$ step-by-step: $$x(x-2)(x+3) = x[(x-2)(x+3)]$$ Expand the inner bracket: $$(x-2)(x+3) = x^2 + 3x - 2x - 6 = x^2 + x - 6$$ Multiply by $$x$$: $$x(x^2 + x - 6) = x^3 + x^2 - 6x$$ 4. **Rewrite the equation with expanded terms:** $$x^3 + x^2 - 6x - 18 = 0$$ 5. **Use substitution to find roots:** We want to find $$x$$ such that: $$x^3 + x^2 - 6x = 18$$ 6. **Graphical interpretation:** Plot the function $$y = x^3 + x^2 - 6x$$ and the horizontal line $$y = 18$$. The roots are the $$x$$-values where these graphs intersect. 7. **Check given options:** - For $$x = -3$$: $$(-3)^3 + (-3)^2 - 6(-3) = -27 + 9 + 18 = 0$$ (not 18) - For $$x = 0$$: $$0 + 0 - 0 = 0$$ (not 18) - For $$x = 2$$: $$8 + 4 - 12 = 0$$ (not 18) - For $$x = 3$$: $$27 + 9 - 18 = 18$$ (matches 18) 8. **Conclusion:** The root of the equation $$x(x-2)(x+3) = 18$$ is $$x = 3$$. **Final answer:** $$\boxed{3}$$