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\left(m+n\right)\left(m^{6}-m^{3}n^{3}+m^{2}n^{4}-mn^{5}+n^{6}+n^{2}m^{4}-nm^{5}\right)
Consider m^{7}+n^{7} as a polynomial over variable m. Find one factor of the form m^{k}+p, where m^{k} divides the monomial with the highest power m^{7} and p divides the constant factor n^{7}. One such factor is m+n. Factor the polynomial by dividing it by this factor.