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2\left(64xy^{3}+27x\right)
Factor out 2.
x\left(64y^{3}+27\right)
Consider 64xy^{3}+27x. Factor out x.
\left(4y+3\right)\left(16y^{2}-12y+9\right)
Consider 64y^{3}+27. Rewrite 64y^{3}+27 as \left(4y\right)^{3}+3^{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).
2x\left(4y+3\right)\left(16y^{2}-12y+9\right)
Rewrite the complete factored expression. Polynomial 16y^{2}-12y+9 is not factored since it does not have any rational roots.