By solving the given quadratic equation, which is X² - 3x² - 4 - 1/(x-1) - 5 = 0, we find the roots of the equation. This may seem a bit complicated at first due to the presence of a fractional term, but with some algebraic manipulation and by using the quadratic formula, we can arrive at an exact solution.
The roots of the equation are the values of 'x' that make the equation equal to zero. These are given by the solution set:
{x = 0.0469846051718295 - 2.10063167857333i, x = 0.0469846051718295 + 2.10063167857333i, x = 0.906030789656341}
The first two solutions are complex numbers, which might not match with the given options. However, the real number solution, x = 0.906030789656341, seems to approximately match with choice a) x = 0.80 and x = 0.91.
The solution might remain approximate due to round-off or computational issues in the calculation process. Therefore, it's good always to keep in mind that these are approximated solutions of our original quadratic equation.
So, looking at these solutions, we can conclude that none of the provided options exactly match our solutions. However, the nearest possible answer seems to be (a), given the small error margin due to possible rounding issues.