Factoring and Simplifying the Expression (x4)^2(x+4)(x4)(x+4)^2+3(x^216)
This article will guide you through the process of factoring and simplifying the expression (x4)^2(x+4)(x4)(x+4)^2+3(x^216).
Recognizing the Pattern
The expression contains repeated terms and the difference of squares pattern. Let's break it down:
 (x4)^2(x+4): This term has (x4) as a common factor.
 (x4)(x+4)^2: This term also has (x4) as a common factor.
 3(x^216): This term represents the difference of squares: (x+4)(x4)
Factoring the Expression

Factor out the common factor (x4):
(x4)^2(x+4)  (x4)(x+4)^2 + 3(x^216) = (x4)[(x4)(x+4)(x+4)^2 + 3(x+4)(x4)]

Factor out the common factor (x+4) from the remaining expression:
(x4)[(x4)(x+4)(x+4)^2 + 3(x+4)(x4)] = (x4)(x+4)[(x4)(x+4) + 3(x4)]

Simplify the expression within the brackets:
(x4)(x+4)[(x4)(x+4) + 3(x4)] = (x4)(x+4)[3x  12]

Factor out the common factor 3 from the last term:
(x4)(x+4)[3x  12] = 3(x4)(x+4)(x4)
Final Simplified Expression
Therefore, the simplified form of the expression is 3(x4)^2(x+4).
This process demonstrates how recognizing patterns and factoring can significantly simplify complex algebraic expressions.