To factor the expression \( 36u^2 - 64 \), we can first recognize that it is a difference of squares. The expression can be written as:
\[ 36u^2 - 64 = (6u)^2 - 8^2 \]
We can then apply the difference of squares formula, which states that \( a^2 - b^2 = (a - b)(a + b) \).
Here, \( a = 6u \) and \( b = 8 \). Therefore, we can factor the expression as follows:
\[ 36u^2 - 64 = (6u - 8)(6u + 8) \]
Next, we can simplify the factors further. We notice that both \( 6u - 8 \) and \( 6u + 8 \) have a common factor of 2.
Factoring out 2 gives:
\[ 6u - 8 = 2(3u - 4) \] \[ 6u + 8 = 2(3u + 4) \]
Now substituting these back into our expression gives:
\[ 36u^2 - 64 = (2(3u - 4))(2(3u + 4)) \]
This simplifies to:
\[ = 4(3u - 4)(3u + 4) \]
Thus, the completely factored form of \( 36u^2 - 64 \) is:
\[ \boxed{4(3u - 4)(3u + 4)} \]