Simplifying the Expression (x+4)(x^24x+16)(64x^3)
This article will explore how to simplify the expression (x+4)(x^24x+16)(64x^3). We'll delve into the process, stepbystep, and explain the reasoning behind each action.
Understanding the Expression
At first glance, the expression might seem complex. However, it's built from recognizable patterns that we can exploit for simplification.
 The first part (x+4)(x^24x+16): This resembles the pattern of a sum of cubes, where (a+b)(a^2ab+b^2) = a^3 + b^3.
 The second part (64x^3): This is a difference of cubes pattern, where (a^3  b^3) = (ab)(a^2 + ab + b^2).
Simplifying using the Patterns
Let's apply these patterns to our expression:

Identify 'a' and 'b':
 In the first part, a = x and b = 4.
 In the second part, a = 4 and b = x.

Apply the patterns:
 The first part becomes: x^3 + 4^3 = x^3 + 64.
 The second part becomes: (4x)(4^2 + 4x + x^2) = (4x)(16 + 4x + x^2).

Rewrite the expression: The complete expression now looks like this: x^3 + 64  (4x)(16 + 4x + x^2).

Simplify further:
 Distribute the negative sign: x^3 + 64  64  4x^2  x^3 + 16x + 4x^2.
 Combine like terms: 16x.
Final Result
By applying the patterns of sum and difference of cubes and simplifying the expression, we arrive at the final answer: 16x.