To simplify the equation, we need to first combine like terms:

x² + 2x + √(x²) + 2x + 8 = 12
x² + 2x + x + 2x + 8 = 12
x² + 5x + 8 = 12

Now, to solve for x, we need to move all terms to one side of the equation:

x² + 5x + 8 - 12 = 0
x² + 5x - 4 = 0

Now, we have a quadratic equation in standard form. We can solve this equation by factoring or using the quadratic formula. Let's use the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

In this case, a = 1, b = 5, and c = -4:

x = [-5 ± √(5² - 41(-4))] / 2*1
x = [-5 ± √(25 + 16)] / 2
x = [-5 ± √41] / 2

Therefore, the solutions for x are:

x = (-5 + √41) / 2
x = (-5 - √41) / 2