To solve the equation x^2 (2+√3) - (√3+1)x - 1 = 0, we can use the quadratic formula.

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 2+√3, b = -(√3+1), and c = -1. Plugging these values into the quadratic formula, we get:

x = (-(√3+1) ± √((√3+1)^2 - 4(2+√3)(-1))) / (2(2+√3))

Simplifying the expression under the square root:

x = (-(√3+1) ± √(3+2√3+1 - 4(2+√3)(-1))) / (2(2+√3))
x = (-(√3+1) ± √(4+2√3+4+8+4√3)) / (2(2+√3))
x = (-(√3+1) ± √(16+6√3)) / (2(2+√3))

Now, we can simplify further by factoring out a 2 from the numerator:

x = -((√3+1) ± √(16+6√3)) / (4+2√3)

Finally, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 4-2√3:

x = -((√3+1) ± √(16+6√3)) / (4+2√3) * (4-2√3)/(4-2√3)
x = -((√3+1)(4-2√3) ± √(16+6√3)(4-2√3)) / (16-12)
x = -((4√3-2√3+4-2√3) ± √(64-24)) / 4
x = -(2√3+4) ± √40 / 4
x = (-2√3-4 ± 2√10) / 4
x = -√3/2 - 1 ± √10/2

Therefore, the solutions to the equation are:

x = -√3/2 - 1 + √10/2
x = -√3/2 - 1 - √10/2