Let's solve for x in the equation 3x(x-1) - 17 = x(1+3x) + 1:

Expand both sides of the equation:
3x^2 - 3x - 17 = x + 3x^2 + 1

Combine like terms:
-3x - 17 = 4x^2 + x + 1

Rearrange the terms:
0 = 4x^2 + 4x + 17

Now, let's solve for x by setting the equation equal to 0:
4x^2 + 4x + 17 = 0

Since this is a quadratic equation, we can use the quadratic formula to solve for x:
x = [-b ± √(b^2 - 4ac)] / 2a

In our case, a = 4, b = 4, and c = 17. Plugging in these values, we get:
x = [-4 ± √(4^2 - 4417)] / 2*4
x = [-4 ± √(16 - 272)] / 8
x = [-4 ± √(-256)] / 8
x = [-4 ± 16i] / 8
x = -1/2 ± 2i

Therefore, the solutions for x are x = -1/2 + 2i and x = -1/2 - 2i.