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9\left(a^{8}-9\right)
Factor out 9.
\left(a^{4}-3\right)\left(a^{4}+3\right)
Consider a^{8}-9. Rewrite a^{8}-9 as \left(a^{4}\right)^{2}-3^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).
9\left(a^{4}-3\right)\left(a^{4}+3\right)
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: a^{4}-3,a^{4}+3.