Paragon: Equations of this kind can't be solved exactly, i.e the solutions can't be expressed in elementary functions. You can use a numerical method, such as Newton's Method, to approximate the solution.
Yes, you seem to believe that
x^2\frac{A}{x^2} = \frac{A}{x^4}
which is not true.
In fact x^2\frac{A}{x^2} = A
So
(x^2-A/x^2)(x^2-A/x^2) = x^4 - A - A + A^2/X^4
[Edit]: Missed one +-sign.
Just a small error. You forgot to simplify 3x-x to 2x.
A(x^2-1) + B(x-1) + C = (3x-1)(x+1)
<=>
Ax^2-A+Bx-B+C = 3x^2+3x-x-1 = 3x^2+2x-1
<=>
Ax^2+Bx-A-B+C = 3x^2+2x-1
Now try again.