1. **State the problem:** Find the value of the expression $$x^3 - 8y^3 - 36xy - 216$$ when $$x = 2y + 6$$. 2. **Substitute the expression for $$x$$:** Replace $$x$$ with $$2y + 6$$ in the expression: $$ (2y + 6)^3 - 8y^3 - 36(2y + 6)y - 216 $$ 3. **Expand the cube:** Use the binomial expansion formula $$ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$ with $$a = 2y$$ and $$b = 6$$: $$ (2y)^3 + 3(2y)^2(6) + 3(2y)(6)^2 + 6^3 = 8y^3 + 72y^2 + 216y + 216 $$ 4. **Rewrite the expression:** $$ 8y^3 + 72y^2 + 216y + 216 - 8y^3 - 36(2y^2 + 6y) - 216 $$ 5. **Simplify the terms:** - The $$8y^3$$ and $$-8y^3$$ cancel out. - Expand $$-36(2y^2 + 6y) = -72y^2 - 216y$$. So the expression becomes: $$ 72y^2 + 216y + 216 - 72y^2 - 216y - 216 $$ 6. **Combine like terms:** $$ 72y^2 - 72y^2 + 216y - 216y + 216 - 216 = 0 $$ **Final answer:** $$\boxed{0}$$