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