To expand and fully simplify the expression [tex]\(5 b (b + 3) - b (3 b - 2)\)[/tex], we will carry out the following steps:
1. Distribute the multiplication within each term.
2. Combine like terms to simplify the expression.
Let's break it down step by step.
### Step 1: Distribute within each term
First, we'll distribute the terms inside the parentheses for each part of the expression separately.
For the term [tex]\(5 b (b + 3)\)[/tex]:
[tex]\[ 5 b (b + 3) = 5 b \cdot b + 5 b \cdot 3 = 5 b^2 + 15 b \][/tex]
Next, for the term [tex]\(-b (3 b - 2)\)[/tex]:
[tex]\[ -b (3 b - 2) = -b \cdot 3 b - (-b) \cdot 2 = -3 b^2 + 2 b \][/tex]
### Step 2: Combine like terms
Now, we take both expanded parts and combine them:
[tex]\[ 5 b^2 + 15 b - 3 b^2 + 2 b \][/tex]
Group the [tex]\(b^2\)[/tex] terms together and the [tex]\(b\)[/tex] terms together:
[tex]\[ (5 b^2 - 3 b^2) + (15 b + 2 b) \][/tex]
Combine the coefficients of each type of term:
[tex]\[ 2 b^2 + 17 b \][/tex]
So, the fully expanded and simplified expression is:
[tex]\[ \boxed{2 b^2 + 17 b} \][/tex]
