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\left(m-2n\right)\left(m^{2}+2mn+4n^{2}\right)
Rewrite m^{3}-8n^{3} as m^{3}-\left(2n\right)^{3}. The difference of cubes can be factored using the rule: a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right).