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\left(5a+2b\right)\left(25a^{2}-10ab+4b^{2}\right)
Rewrite 125a^{3}+8b^{3} as \left(5a\right)^{3}+\left(2b\right)^{3}. The sum of cubes can be factored using the rule: p^{3}+q^{3}=\left(p+q\right)\left(p^{2}-pq+q^{2}\right).