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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.