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\left(x-4ay\right)\left(x^{2}+4axy+16a^{2}y^{2}\right)
Rewrite x^{3}-64a^{3}y^{3} as x^{3}-\left(4ay\right)^{3}. The difference of cubes can be factored using the rule: p^{3}-q^{3}=\left(p-q\right)\left(p^{2}+pq+q^{2}\right).