Simplify the expression: (b^2 + 2b)^2 - b^2(b - 1)(b + 1) + 2b(3 - 2b^2)

Let's simplify the given expression step by step:

1. Expand the first term:

   \[ (b^2 + 2b)^2 = (b^2)^2 + 2(b^2)(2b) + (2b)^2 = b^4 + 4b^3 + 4b^2 \]

2. Expand the second term:

   \[ -b^2(b - 1)(b + 1) = -b^2(b^2 - 1) = -b^4 + b^2 \]

3. Expand the third term:

   \[ 2b(3 - 2b^2) = 6b - 4b^3 \]

4. Combine all the simplified terms:

   \[ (b^4 + 4b^3 + 4b^2) + (-b^4 + b^2) + (6b - 4b^3) \]

5. Group like terms:

   \[ (b^4 - b^4) + (4b^3 - 4b^3) + (4b^2 + b^2) + 6b \]

6. Simplify:

   \[ 0 + 0 + 5b^2 + 6b = 5b^2 + 6b \]

Answer: $$5b^2 + 6b$$