The problem asks us to factor the expression $256x^4 - 1$ completely.

1. Problem Description

The problem asks us to factor the expression $256x^4 - 1$ completely.

2. Solution Steps

We are given the expression $256x^4 - 1$. This is a difference of squares. We can write it as

$256x^4 - 1 = (16x^2)^2 - (1)^2$.

Using the difference of squares formula $a^2 - b^2 = (a-b)(a+b)$, we have

$256x^4 - 1 = (16x^2 - 1)(16x^2 + 1)$.

Now, we observe that $16x^2 - 1$ is also a difference of squares. We can write it as $(4x)^2 - (1)^2$. Thus, we can factor it as:

$16x^2 - 1 = (4x - 1)(4x + 1)$.

The term $16x^2 + 1$ is a sum of squares and cannot be factored further using real numbers. Therefore, the complete factorization of $256x^4 - 1$ is:

$256x^4 - 1 = (4x - 1)(4x + 1)(16x^2 + 1)$.

3. Final Answer

$(4x - 1)(4x + 1)(16x^2 + 1)$