The correct factorization would be (5x - 4)(25x^2 + 20x + 16).
The law of factoring cubes states that we use the following formula when we have two cubes being subtracted by one another.
(a - b)(a^2 + ab + b^2)
If we take the cube root of 125x^3 we get 5x, which will be our a value. If we take the cube root of 64, we get 4, which will be our b value. Then we plug in to the formula.
(5x - 4)((5x)^2 + (5x)(4) + 4^2)
(5x - 4)(25x^2 + 20x + 16)
I think the problem with your second answer is that you just did not properly multiply the 5x and the 4 to get 20x (instead you got 15x).