To solve the equation, we need to find the common denominator for all the fractions. The common denominator is (3x - 1)(x - 1)(3x^2 - 4x + 1).

Multiplying each term by the common denominator, we get:

[(x - 1)(3x^2 - 4x + 1)] + (2x + 2)(3x - 1) - (3x^2 + 1)(x - 1) = (3x - 1)(x - 1)(3x^2 - 4x + 1)

Expanding each term, we have:

3x^3 - 4x^2 + x - 3x + 4 - 3x^2 + 4x - 1 - 3x^3 + 4x^2 - x + 3x^2 - 4x + 1 = 3x^3 - 4x^2 + x - 3x + 4 - 3x^2 + 4x - x + 3x^2 - 4x + 1

Simplifying, we get:

3x^3 - 4x^2 + x - 3x + 4 - 3x^2 + 4x - 1 - 3x^3 + 4x^2 - x + 3x^2 - 4x + 1 = 0

Combining like terms, we have:

-2x + 9 = 0

Solving for x, we get:

2x = 9
x = 9/2

Therefore, the solution to the equation is x = 9/2.