To solve this equation, we first cross multiply to eliminate the fractions:

(x-1)(4x-1) = (x+2)(2x+12)

Expanding both sides:

4x^2 - x - 4x + 1 = 2x^2 + 24x + 4x + 24

Combining like terms:

4x^2 - 5x + 1 = 2x^2 + 28x + 24

Subtract 2x^2 and 28x from both sides:

2x^2 - 33x + 1 = 24

Rearranging terms:

2x^2 - 33x - 23 = 0

We can solve this quadratic equation using the quadratic formula:

x = (-(-33) ± √((-33)^2 - 4(2)(-23)))/(2(2))
x = (33 ± √(1089 + 184))/4
x = (33 ± √1273)/4

Therefore, the solutions to the equation are:

x = (33 + √1273)/4 or x = (33 - √1273)/4