Factor the expression $-x^3 - 125$.

1. Problem Description

Factor the expression $-x^3 - 125$.

2. Solution Steps

First, factor out a -1 from the expression:

$-x^3 - 125 = -(x^3 + 125)$

We recognize that $x^3 + 125$ is a sum of cubes, where $x^3 = a^3$ and $125 = 5^3$. Therefore, $a = x$ and $b = 5$.

Recall the sum of cubes formula:

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

Applying the sum of cubes formula to $x^3 + 5^3$, we get:

$x^3 + 5^3 = (x+5)(x^2 - 5x + 25)$

Therefore,

$-(x^3 + 125) = -(x+5)(x^2 - 5x + 25) = (-x-5)(x^2 - 5x + 25)$ or $-(x+5)(x^2-5x+25)$

3. Final Answer

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