(x+2)^2(x-1)^2-(x-4)^2(x+4)^2

2 min read Jun 16, 2024

Factoring and Simplifying (x+2)²(x-1)² - (x-4)²(x+4)²

This expression looks complex at first glance, but it can be simplified using the difference of squares pattern and some strategic factoring.

Recognizing the Difference of Squares Pattern

The expression contains two squared terms being subtracted:

(x+2)²(x-1)² is the square of (x+2)(x-1)
(x-4)²(x+4)² is the square of (x-4)(x+4)

The difference of squares pattern states: a² - b² = (a+b)(a-b)

Applying the Pattern

Let's apply this pattern to our expression:

Identify a and b:
- a = (x+2)(x-1)
- b = (x-4)(x+4)
Substitute into the pattern: (x+2)²(x-1)² - (x-4)²(x+4)² = [(x+2)(x-1) + (x-4)(x+4)][(x+2)(x-1) - (x-4)(x+4)]

Expanding and Simplifying

Now, we need to expand the products within the brackets:

(x+2)(x-1) + (x-4)(x+4):
- x² + x - 2 + x² - 16 = 2x² + x - 18
(x+2)(x-1) - (x-4)(x+4):
- x² + x - 2 - (x² - 16) = x + 14

Finally, multiply these two results:

(2x² + x - 18)(x + 14) = 2x³ + 29x² + 106x - 252

Conclusion

By recognizing the difference of squares pattern and carefully expanding and simplifying, we have successfully factored and simplified the original expression, arriving at 2x³ + 29x² + 106x - 252.