1. **State the problem:** Solve the factoring problem $$3x^2 - 18x = -27$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$3x^2 - 18x + 27 = 0$$ 3. **Factor out the greatest common factor (GCF):** The GCF of $$3x^2$$, $$-18x$$, and $$27$$ is 3: $$3x^2 - 18x + 27 = 3(x^2 - 6x + 9)$$ 4. **Factor the quadratic inside the parentheses:** Recognize that $$x^2 - 6x + 9$$ is a perfect square trinomial: $$x^2 - 6x + 9 = (x - 3)^2$$ 5. **Write the factored form:** $$3(x - 3)^2 = 0$$ 6. **Solve for $$x$$:** Set the factor equal to zero: $$3(x - 3)^2 = 0 \implies (x - 3)^2 = 0$$ 7. **Find the root:** $$x - 3 = 0 \implies x = 3$$ **Final answer:** $$x = 3$$