We are asked to factor the expression $125x^3 - 1$ using the difference of cubes formula.

1. Problem Description

We are asked to factor the expression $125x^3 - 1$ using the difference of cubes formula.

2. Solution Steps

The difference of cubes formula is:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

We can rewrite the given expression $125x^3 - 1$ as $(5x)^3 - 1^3$.

Here, $a = 5x$ and $b = 1$.

Applying the difference of cubes formula:

$(5x)^3 - 1^3 = (5x - 1)((5x)^2 + (5x)(1) + 1^2)$

$= (5x - 1)(25x^2 + 5x + 1)$

3. Final Answer

$(5x - 1)(25x^2 + 5x + 1)$