1. **Stating the problem:** Simplify and factor the expression $a(a + 1) - (a + 1)^2$. 2. **Formula and rules:** Use distributive property $x(y + z) = xy + xz$ and factoring techniques. 3. **Expand each term:** $$a(a + 1) = a^2 + a$$ $$(a + 1)^2 = (a + 1)(a + 1) = a^2 + 2a + 1$$ 4. **Rewrite the expression:** $$a^2 + a - (a^2 + 2a + 1)$$ 5. **Distribute the minus sign:** $$a^2 + a - a^2 - 2a - 1$$ 6. **Combine like terms:** $$\cancel{a^2} + a - \cancel{a^2} - 2a - 1 = (a - 2a) - 1 = -a - 1$$ 7. **Factor out the negative sign:** $$- (a + 1)$$ **Final answer:** $$- (a + 1)$$