Rewrite $a^{4}-1$ as $\left(a^{2}\right)^{2}-1^{2}$. The difference of squares can be factored using the rule: $p^{2}-q^{2}=\left(p-q\right)\left(p+q\right)$.
$$\left(a^{2}-1\right)\left(a^{2}+1\right)$$
Consider $a^{2}-1$. Rewrite $a^{2}-1$ as $a^{2}-1^{2}$. The difference of squares can be factored using the rule: $p^{2}-q^{2}=\left(p-q\right)\left(p+q\right)$.
$$\left(a-1\right)\left(a+1\right)$$
Rewrite the complete factored expression. Polynomial $a^{2}+1$ is not factored since it does not have any rational roots.
