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\left(3a^{2}-4\right)\left(3a^{2}+4\right)
Rewrite 9a^{4}-16 as \left(3a^{2}\right)^{2}-4^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right). The following polynomials are not factored since they do not have any rational roots: 3a^{2}-4,3a^{2}+4.