1. **State the problem:** Simplify the expression $X - 2 (x + 2) - 2 (x - 2)(x - 2)$. 2. **Rewrite the expression:** $$X - 2(x + 2) - 2(x - 2)^2$$ 3. **Expand the terms:** - Expand $-2(x + 2)$: $$-2 \times x - 2 \times 2 = -2x - 4$$ - Expand $(x - 2)^2$ using the formula $(a - b)^2 = a^2 - 2ab + b^2$: $$x^2 - 2 \times x \times 2 + 2^2 = x^2 - 4x + 4$$ - Multiply by $-2$: $$-2(x^2 - 4x + 4) = -2x^2 + 8x - 8$$ 4. **Substitute expansions back:** $$X - 2x - 4 - 2x^2 + 8x - 8$$ 5. **Combine like terms:** - Combine $X$ and $-2x$ and $8x$: $$X + (-2x + 8x) = X + 6x$$ - Combine constants $-4$ and $-8$: $$-4 - 8 = -12$$ 6. **Final simplified expression:** $$-2x^2 + X + 6x - 12$$ **Answer:** $$\boxed{-2x^2 + X + 6x - 12}$$