The given expression to simplify is:

$\frac{4(4x - 5)^{1/5} - (x - 4)(4x - 5)^{-4/5}}{(4x - 5)^{4/5}}.$

Step 1: Factor out common terms in the numerator

To simplify, first factor out the common term $(4x - 5)^{-4/5}$ from the numerator.

$4(4x - 5)^{1/5} = 4(4x - 5)^{5/5 \times 1/5} = 4(4x - 5)^{-4/5} \times (4x - 5).$

$(x - 4)(4x - 5)^{-4/5} = (x - 4)(4x - 5)^{-4/5}.$

Now, the numerator can be rewritten as:

$(4(4x - 5) - (x - 4))(4x - 5)^{-4/5}$

Step 2: Simplify the fraction

The expression now becomes:

$\frac{4(4x - 5) - (x - 4)}{(4x - 5)^{4/5}}.$

Since the common factor $(4x - 5)^{4/5}$ divides out with $(4x - 5)$ resulting in:

Step 3: Simplify numerator

The numerator simplifies to:

$\frac{4(4x - 5)(-x-4)}{(4x-5)^1} = x+4$

We simplify to expression, the result turns into [(x-4).