don't expand! it is quadratic in (x - 1)/x

replace (x - 1)/x with another variable, say y, and you will see that

y^2 - (77/12)y + 10 = 0

solve for the roots of that equation, when you get them, replace y with (x - 1)/x and solve for x

EDIT: Oh, it was x - 1/x not (x - 1)/x. that does not change the process though

EDIT 2: by the way, you expanded incorrectly

EDIT 3: I must be tired. the expansion was correct, so scratch out EDIT 2. I still think my original plan is best though. Yes you have to do the quadratic formula 2 or 3 times (2 if you use Plato's method), but to that's easier than working with the quartic equation you would get if you continued down the path you were going