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**Soroban** Hello, bilbobaggins!

Factor a "minus" out of the middle denominator . . .

We have: .$\displaystyle \frac{4x}{x-1} - \frac{4}{x^2-1} - \frac{2}{x+1}$

Factor: .$\displaystyle \frac{4x}{x-1} - \frac{4}{(x-1)(x+1)} - \frac{2}{x+1}\quad\hdots\;\;\text{LCD} \:=\: (x-1)(x+1)$

Convert: .$\displaystyle \frac{4x}{x-1}\cdot{\color{blue}\frac{x+1}{x+1}} - \frac{4}{(x-1)(x+1)} - \frac{2}{x+1}\cdot{\color{blue}\frac{x-1}{x-1}} $

. . $\displaystyle = \;\frac{4x(x+1) - 4 - 2(x-1)}{(x-1)(x+1)} \;=\;\frac{4x^2+4x-4-2x+2}{(x-1)(x+1)} \;=\;\frac{4x^2+2x-2}{(x-1)(x+1)}$

Factor: .$\displaystyle \frac{2(2x^2+x-2)}{(x-1)(x+1)} \;=\;\frac{2(2x-1)(x+1)}{(x-1)(x+1)} \;=\; \frac{2(2x-1)}{x-1} $