\(\dfrac{-x^2-1}{x\left(x-1\right)}=\dfrac{-\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)}=\dfrac{-x-1}{x}\)
\(\dfrac{\left(1-x\right)\left(1+x\right)}{x\left(x-1\right)}\\ \dfrac{\left(1-x\right)\left(1+x\right)}{-x\left(1-x\right)}\\ -\dfrac{1+x}{x}\)
(1-x)(1+x) / x(x-1) = -(x-1)(1+x) / x(x-1) = -(1+x) / x