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

(2|x|-3)² - |x| - 6 / (4x - 1) = 0

Let's make a substitution: Let y = |x|, so we have:

(2y-3)² - y - 6 / (4x - 1) = 0

Expanding the squared term:

(4y² - 12y + 9) - y - 6 / (4x - 1) = 0

Combining like terms:

4y² - 13y + 3 / (4x - 1) = 0

Let's now solve for y by setting the numerator equal to zero:

4y² - 13y + 3 = 0

This is a quadratic equation that can be solved using the quadratic formula:

y = (13 ± √(13² - 443))/(2*4)

y = (13 ± √(169 - 48))/8

y = (13 ± √121)/8

y = (13 ± 11)/8

y = 24/8 or y = 2/8

y = 3 or y = 0.25

Now substitute back to find possible values of x:

x = 3 or x = -3

Let's check our solutions:

For x = 3:
(23-3)² - |3| - 6 / (43 - 1) = 0
(6-3)² - 3 - 6 / (12-1) = 0
3² - 3 - 6 / 11 = 0
9 - 3 - 6 / 11 = 0
0 / 11 = 0 -> True

For x = -3:
(2-3-3)² - |-3| - 6 / (4-3 - 1) = 0
(-6-3)² - 3 - 6 / (-12-1) = 0
-9² - 3 - 6 / -13 = 0
81 - 3 - 6 / -13 = 0
72 - 6 / -13 = 0
66 / -13 ≠ 0 -> False

Therefore, x = 3 is the solution to the equation.