The problem is to factor the polynomial $4x^3 - 8x^2 - 36x - 72$.

1. Problem Description

The problem is to factor the polynomial $4x^3 - 8x^2 - 36x - 72$.

2. Solution Steps

Step 1: Factor out the greatest common factor (GCF) from all terms.

The GCF of the coefficients 4, -8, -36, and -72 is

4. So, we factor out 4 from the polynomial.

$4x^3 - 8x^2 - 36x - 72 = 4(x^3 - 2x^2 - 9x - 18)$.

Step 2: Factor by grouping.

We group the terms in pairs: $(x^3 - 2x^2)$ and $(-9x - 18)$.

From the first group, we can factor out $x^2$, and from the second group, we can factor out $-9$.

$x^3 - 2x^2 - 9x - 18 = x^2(x - 2) - 9(x + 2)$.

I believe there is a typo. The problem should be $4x^3 - 8x^2 - 36x + 72$. Let's resolve the question with that assumption:

The problem is to factor the polynomial $4x^3 - 8x^2 - 36x + 72$.

Step 1: Factor out the greatest common factor (GCF) from all terms.

The GCF of the coefficients 4, -8, -36, and 72 is

4. So, we factor out 4 from the polynomial.

$4x^3 - 8x^2 - 36x + 72 = 4(x^3 - 2x^2 - 9x + 18)$.

Step 2: Factor by grouping.

We group the terms in pairs: $(x^3 - 2x^2)$ and $(-9x + 18)$.

From the first group, we can factor out $x^2$, and from the second group, we can factor out $-9$.

$x^3 - 2x^2 - 9x + 18 = x^2(x - 2) - 9(x - 2)$.

Step 3: Factor out the common binomial factor.

We see that $(x - 2)$ is a common factor.

$x^2(x - 2) - 9(x - 2) = (x - 2)(x^2 - 9)$.

Step 4: Factor the difference of squares.

We recognize that $x^2 - 9$ is a difference of squares, $x^2 - 3^2$, which factors as $(x - 3)(x + 3)$.

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

Step 5: Combine all factors.

$4x^3 - 8x^2 - 36x + 72 = 4(x - 2)(x - 3)(x + 3)$.

3. Final Answer

$4(x-2)(x-3)(x+3)$