The problem asks to factor out the greatest common factor from the expression $4a^2b^3 - 8a^3b^3 + 4a^3b^2 - 36a^4b^2$.

1. Problem Description

The problem asks to factor out the greatest common factor from the expression $4a^2b^3 - 8a^3b^3 + 4a^3b^2 - 36a^4b^2$.

2. Solution Steps

First, identify the coefficients of each term: 4, -8, 4, and -

3

6. The greatest common factor of these coefficients is

4.

Next, identify the powers of $a$ in each term: $a^2$, $a^3$, $a^3$, and $a^4$.

The lowest power of $a$ is $a^2$.

Then, identify the powers of $b$ in each term: $b^3$, $b^3$, $b^2$, and $b^2$.

The lowest power of $b$ is $b^2$.

Therefore, the greatest common factor of the entire expression is $4a^2b^2$.

Now, factor out $4a^2b^2$ from each term:

$4a^2b^3 = 4a^2b^2(b)$

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

$4a^3b^2 = 4a^2b^2(a)$

$-36a^4b^2 = 4a^2b^2(-9a^2)$

So, we can rewrite the expression as:

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

Finally, factor out the greatest common factor $4a^2b^2$ from the entire expression:

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

3. Final Answer

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