Solution:
First, we need to combine the fractions on the left side of the equation by finding their lowest common denominator, which is (2x-1)(2x^2+x-1). This gives us:
(2x^2+x-1) + 4(2x^2+x-1) = 4(2x-1)(2x^2+x-1)
Next, we can distribute the 4 on the right side and simplify the equations:
2x^2 + x - 1 + 8x^2 + 4x - 4 = 8x^3 - 4x^2 + 4x - 4
Then, we can combine like terms on both sides:
10x^2 + 5x - 5 = 8x^3 - 4x^2 + 4x - 4
Now, we can move all the terms to one side, leaving 0 on the other:
8x^3 - 14x^2 + x - 1 = 0
This equation can be solved using the Rational Root Theorem, which states that any rational root of a polynomial is in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Using this knowledge, we can find that the rational roots of this equation are -1/2 and 1/4.
We can then use synthetic division to find that -1/2 is a root, leaving us with a new equation:
(2x - 1)(8x^2 - 19x + 2) = 0
From this, we know that 2x - 1 = 0 or 8x^2 - 19x + 2 = 0. Solving these equations gives us the solutions x = 1/2 and x = 1/8. Therefore, the solution to the given equation is x = 1/2 or x = 1/8.