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z\left(x-1\right)-\left(x-1\right)
Do the grouping zx-z-x+1=\left(zx-z\right)+\left(-x+1\right), and factor out z in the first and -1 in the second group.
\left(x-1\right)\left(z-1\right)
Factor out common term x-1 by using distributive property.