The problem asks us to factor out the greatest common factor from the expression $10ab^2 + 5ab - 15a^3b$.

1. Problem Description

The problem asks us to factor out the greatest common factor from the expression $10ab^2 + 5ab - 15a^3b$.

2. Solution Steps

First, we identify the terms in the expression: $10ab^2$, $5ab$, and $-15a^3b$.

Next, we find the greatest common factor (GCF) of the coefficients: 10, 5, and -

1

5. The GCF is

5.

Now, we find the GCF of the variable factors.

The variable factors are $a$, $b^2$; $a$, $b$; and $a^3$, $b$.

The common variable factors are $a$ and $b$. The lowest power of $a$ appearing in the terms is $a^1 = a$, and the lowest power of $b$ is $b^1 = b$. Therefore the GCF of the variable factors is $ab$.

The GCF of the entire expression is $5ab$.

Now, we factor out the GCF from each term:

$10ab^2 = 5ab(2b)$

$5ab = 5ab(1)$

$-15a^3b = 5ab(-3a^2)$

Therefore, we have $10ab^2 + 5ab - 15a^3b = 5ab(2b) + 5ab(1) + 5ab(-3a^2) = 5ab(2b + 1 - 3a^2)$.

3. Final Answer

$5ab(2b+1-3a^2)$