1. **State the problem:** Simplify the expression $$(a-1)^2 - 2(a-1)(a+1) + (a+1)^2$$. 2. **Recall the formula:** The square of a binomial is $$(x \, \pm \, y)^2 = x^2 \, \pm \, 2xy \, + \, y^2$$. 3. **Expand each term:** - $$(a-1)^2 = a^2 - 2a + 1$$ - $$(a+1)^2 = a^2 + 2a + 1$$ - $$-2(a-1)(a+1) = -2(a^2 - 1)$$ because $$(a-1)(a+1) = a^2 - 1$$. 4. **Substitute expansions back:** $$a^2 - 2a + 1 - 2(a^2 - 1) + a^2 + 2a + 1$$ 5. **Distribute the -2:** $$a^2 - 2a + 1 - 2a^2 + 2 + a^2 + 2a + 1$$ 6. **Combine like terms:** - Combine $$a^2$$ terms: $$a^2 - 2a^2 + a^2 = \cancel{a^2} - 2a^2 + \cancel{a^2} = 0$$ - Combine $$a$$ terms: $$-2a + 2a = 0$$ - Combine constants: $$1 + 2 + 1 = 4$$ 7. **Final simplified expression:** $$4$$ **Answer:** The expression simplifies to $$4$$.