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\left(a^{2}-4b\right)\left(a^{2}+4b\right)
Rewrite a^{4}-16b^{2} as \left(a^{2}\right)^{2}-\left(4b\right)^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).