1. **State the problem:** Factor the expression $$27x^3 - 125$$. 2. **Recognize the form:** This is a difference of cubes since $$27x^3 = (3x)^3$$ and $$125 = 5^3$$. 3. **Recall the difference of cubes formula:** $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ 4. **Apply the formula:** Let $$a = 3x$$ and $$b = 5$$. 5. Substitute into the formula: $$27x^3 - 125 = (3x - 5)((3x)^2 + (3x)(5) + 5^2)$$ 6. Simplify inside the second parentheses: $$= (3x - 5)(9x^2 + 15x + 25)$$ 7. **Final factored form:** $$27x^3 - 125 = (3x - 5)(9x^2 + 15x + 25)$$