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x^{2}+\left(-2+m\right)x-m+1
Consider x^{2}-2x+mx-m+1 as a polynomial over variable x.
\left(x-1\right)\left(x+m-1\right)
Find one factor of the form x^{k}+n, where x^{k} divides the monomial with the highest power x^{2} and n divides the constant factor -m+1. One such factor is x-1. Factor the polynomial by dividing it by this factor.