Consider $a^{4}+a^{2}-2$. Find one factor of the form $a^{k}+m$, where $a^{k}$ divides the monomial with the highest power $a^{4}$ and $m$ divides the constant factor $-2$. One such factor is $a^{2}+2$. Factor the polynomial by dividing it by this factor.
$$\left(a^{2}+2\right)\left(a^{2}-1\right)$$
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 $a^{2}+2$ is not factored since it does not have any rational roots.
