The given expression is:
6 + 3 = 2 * (4x^2 - 1) + 2x + 1 + 2x - 1 + 1

First, simplify the expression within the parenthesis:

4x^2 - 1: This is a difference of squares and can be factored as (2x + 1)(2x - 1)

Now the expression becomes:
6 + 3 = 2 * (2x + 1)(2x - 1) + 2x + 1 + 2x - 1 + 1

Now distribute the 2:
6 + 3 = 4x^2 - 4 + 2x + 1 + 2x - 1 + 1

Combine like terms:
6 + 3 = 4x^2 + 4x

So the final simplified expression is:
9 = 4x^2 + 4x