The problem asks to factor out the greatest common factor from the expression $2a^2b^3 - 18a^3b^3 + 2a^3b^2 - 8a^4b^2$.

1. Problem Description

The problem asks to factor out the greatest common factor from the expression $2a^2b^3 - 18a^3b^3 + 2a^3b^2 - 8a^4b^2$.

2. Solution Steps

First, identify the greatest common factor (GCF) of the coefficients. The coefficients are 2, -18, 2, and -

8. The GCF of these numbers is

2.

Next, find the GCF of the variable $a$. The terms are $a^2$, $a^3$, $a^3$, and $a^4$. The smallest exponent of $a$ is 2, so the GCF for $a$ is $a^2$.

Then, find the GCF of the variable $b$. The terms are $b^3$, $b^3$, $b^2$, and $b^2$. The smallest exponent of $b$ is 2, so the GCF for $b$ is $b^2$.

Therefore, the greatest common factor of the expression is $2a^2b^2$.

Now, factor out the GCF from each term:

$2a^2b^3 = 2a^2b^2(b)$

$-18a^3b^3 = 2a^2b^2(-9ab)$

$2a^3b^2 = 2a^2b^2(a)$

$-8a^4b^2 = 2a^2b^2(-4a^2)$

Combine the factored terms:

$2a^2b^3 - 18a^3b^3 + 2a^3b^2 - 8a^4b^2 = 2a^2b^2(b - 9ab + a - 4a^2)$

3. Final Answer

$2a^2b^2(b - 9ab + a - 4a^2)$