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\left(xy+8\right)\left(x^{2}y^{2}-8xy+64\right)
Rewrite x^{3}y^{3}+512 as \left(xy\right)^{3}+8^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).