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\frac{x+1}{\left(x-1\right)\left(-x-1\right)}
Factor the expressions that are not already factored.
\frac{-\left(-x-1\right)}{\left(x-1\right)\left(-x-1\right)}
Extract the negative sign in 1+x.
\frac{-1}{x-1}
Cancel out -x-1 in both numerator and denominator.
\frac{x+1}{\left(x-1\right)\left(-x-1\right)}
Factor the expressions that are not already factored.
\frac{-\left(-x-1\right)}{\left(x-1\right)\left(-x-1\right)}
Extract the negative sign in 1+x.
\frac{-1}{x-1}
Cancel out -x-1 in both numerator and denominator.