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\left(a-8b^{2}\right)\left(a+8b^{2}\right)
Rewrite a^{2}-64b^{4} as a^{2}-\left(8b^{2}\right)^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).