The problem asks to factor the trinomial $x^3 + x^2 - 72x$ completely. First, find the greatest common factor (GCF) of the terms and factor it out. Then, factor the remaining trinomial.

1. Problem Description

The problem asks to factor the trinomial $x^3 + x^2 - 72x$ completely. First, find the greatest common factor (GCF) of the terms and factor it out. Then, factor the remaining trinomial.

2. Solution Steps

Step 1: Identify the GCF.

The terms are $x^3$, $x^2$, and $-72x$. The GCF is $x$.

Step 2: Factor out the GCF.

$x^3 + x^2 - 72x = x(x^2 + x - 72)$.

Step 3: Factor the remaining trinomial $x^2 + x - 72$.

We are looking for two numbers that multiply to $-72$ and add up to $1$. These numbers are $9$ and $-8$. Therefore,

$x^2 + x - 72 = (x + 9)(x - 8)$.

Step 4: Write the completely factored expression.

$x^3 + x^2 - 72x = x(x^2 + x - 72) = x(x + 9)(x - 8)$.

3. Final Answer

$x(x+9)(x-8)$