Factoring and Simplifying (x+2)²(x1)²  (x4)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(x1)² is the square of (x+2)(x1)
 (x4)²(x+4)² is the square of (x4)(x+4)
The difference of squares pattern states: a²  b² = (a+b)(ab)
Applying the Pattern
Let's apply this pattern to our expression:

Identify a and b:
 a = (x+2)(x1)
 b = (x4)(x+4)

Substitute into the pattern: (x+2)²(x1)²  (x4)²(x+4)² = [(x+2)(x1) + (x4)(x+4)][(x+2)(x1)  (x4)(x+4)]
Expanding and Simplifying
Now, we need to expand the products within the brackets:
 (x+2)(x1) + (x4)(x+4):
 x² + x  2 + x²  16 = 2x² + x  18
 (x+2)(x1)  (x4)(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.