To solve this equation, we need to first get rid of the fractions by finding a common denominator.

Multiply both sides of the equation by (x+2)(x-2) to get rid of the fractions:

[(3x+1)(x-2)] = (x+2)(x-2) + (x-1)(x+2)

Expand on both sides:

3x^2 - 6x + x - 2 = x^2 - 2x + 2x - 4 + x^2 + 2x - 2

Combine like terms:

3x^2 - 5x - 2 = x^2 + 2x - 4 + x^2 + 2x - 2

Rearrange the equation:

3x^2 - 5x - 2 = 2x^2 + 4x - 6

Move all terms to one side to form a quadratic equation:

3x^2 - 5x - 2 - 2x^2 - 4x + 6 = 0

x^2 - 9x + 4 = 0

Now, to solve this quadratic equation, we can use the quadratic formula:

x = (-(-9) ± √((-9)^2 - 414)) / 2*1
x = (9 ± √(81 - 16)) / 2
x = (9 ± √65) / 2

So the solutions to the equation are:

x = (9 + √65) / 2 and x = (9 - √65) / 2