## Simplifying the Expression (a+1)²-(a-1)²-3(a+1)(a-1)

This article will explore the simplification of the algebraic expression **(a+1)²-(a-1)²-3(a+1)(a-1)**. We will utilize algebraic identities and basic arithmetic operations to achieve a simplified form.

### Understanding the Expression

The expression involves three terms:

**(a+1)²:**This is a perfect square trinomial, expanding to**a² + 2a + 1****(a-1)²:**This is also a perfect square trinomial, expanding to**a² - 2a + 1****3(a+1)(a-1):**This represents the product of three factors, where (a+1) and (a-1) form the difference of squares pattern.

### Simplifying Using Identities

Let's utilize the following algebraic identities to simplify the expression:

**Difference of Squares:**(x+y)(x-y) = x² - y²**Perfect Square Trinomial:**(x+y)² = x² + 2xy + y²**Perfect Square Trinomial:**(x-y)² = x² - 2xy + y²

**Step 1:** Expanding the squares

- (a+1)² = a² + 2a + 1
- (a-1)² = a² - 2a + 1

**Step 2:** Applying the Difference of Squares Identity

- 3(a+1)(a-1) = 3(a² - 1²) = 3a² - 3

**Step 3:** Substituting the expanded terms into the original expression

- (a² + 2a + 1) - (a² - 2a + 1) - (3a² - 3)

**Step 4:** Simplifying by combining like terms

- a² + 2a + 1 - a² + 2a - 1 - 3a² + 3
**-3a² + 4a + 3**

### Conclusion

Therefore, the simplified form of the expression **(a+1)²-(a-1)²-3(a+1)(a-1)** is **-3a² + 4a + 3**. This process demonstrates how understanding algebraic identities can significantly simplify complex expressions.