1. **State the problem:** Simplify the expression $X^2 - a^2 - y^2 - 2ay$. 2. **Recall the formula:** Recognize that $y^2 + 2ay + a^2$ is a perfect square trinomial equal to $(y + a)^2$. 3. **Rewrite the expression:** Group terms to see the pattern: $$X^2 - a^2 - y^2 - 2ay = X^2 - (a^2 + y^2 + 2ay)$$ 4. **Use the perfect square identity:** Replace $a^2 + y^2 + 2ay$ with $(y + a)^2$: $$X^2 - (y + a)^2$$ 5. **Apply the difference of squares formula:** $$X^2 - (y + a)^2 = (X - (y + a))(X + (y + a))$$ 6. **Simplify the factors:** $$(X - y - a)(X + y + a)$$ **Final answer:** $$(X - y - a)(X + y + a)$$