To solve this equation, we first need to simplify it:

2(x^2 - 2x) + √(x^2 - 2x - 3) = 9
2x^2 - 4x + √(x^2 - 2x - 3) = 9

Now, let's isolate the square root term by moving the other terms to the other side of the equation:

√(x^2 - 2x - 3) = 9 - 2x^2 + 4x

Next, square both sides of the equation to eliminate the square root:

(x^2 - 2x - 3) = (9 - 2x^2 + 4x)^2
x^2 - 2x - 3 = 81 - 36x^2 + 36x + 4x^2 - 36x
x^2 - 2x - 3 = 81 - 32x^2

Rearrange the equation to set it equal to zero:

32x^2 - x^2 + 2x - 36x - 3 - 81 = 0
31x^2 - 34x - 84 = 0

Now, we have a quadratic equation that can be solved using the quadratic formula:

x = (-(-34) ± √((-34)^2 - 4*31*(-84))) / (2*31)
x = (34 ± √(1156 + 10464)) / 62
x = (34 ± √12020) / 62
x = (34 ± 109.6) / 62

Therefore, the solutions for x are:

x = (34 + 109.6) / 62 = 143.6 / 62 ≈ 2.316
x = (34 - 109.6) / 62 = -75.6 / 62 ≈ -1.219

So, the solutions for the equation are x ≈ 2.316 and x ≈ -1.219.