## Factoring and Simplifying the Expression: (x^2+x-1)^2-(x^2+2x+3)^2

This article explores the process of simplifying the expression **(x^2+x-1)^2-(x^2+2x+3)^2**. We'll use the difference of squares pattern and factoring to arrive at a simplified solution.

### Difference of Squares Pattern

The key to simplifying this expression lies in recognizing the **difference of squares pattern:**

**a² - b² = (a + b)(a - b)**

In our expression, we can identify **a** and **b** as follows:

**a = (x² + x - 1)****b = (x² + 2x + 3)**

### Applying the Pattern

Applying the difference of squares pattern to our expression, we get:

**(x² + x - 1)² - (x² + 2x + 3)² = [(x² + x - 1) + (x² + 2x + 3)][(x² + x - 1) - (x² + 2x + 3)]**

### Simplifying the Expression

Now, we need to simplify the expressions within the brackets:

**[(x² + x - 1) + (x² + 2x + 3)] = (2x² + 3x + 2)****[(x² + x - 1) - (x² + 2x + 3)] = (-x - 4)**

Therefore, the simplified form of the expression becomes:

**(x² + x - 1)² - (x² + 2x + 3)² = (2x² + 3x + 2)(-x - 4)**

### Conclusion

By recognizing the difference of squares pattern and applying it to the given expression, we successfully simplified it to **(2x² + 3x + 2)(-x - 4)**. This process demonstrates the importance of understanding algebraic patterns and their application in simplifying complex expressions.