Find one factor of the form $x^{k}+m$, where $x^{k}$ divides the monomial with the highest power $x^{8}$ and $m$ divides the constant factor $-2$. One such factor is $x^{4}+2$. Factor the polynomial by dividing it by this factor.
$$\left(x^{4}+2\right)\left(x^{4}-1\right)$$
Consider $x^{4}-1$. Rewrite $x^{4}-1$ as $\left(x^{2}\right)^{2}-1^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x^{2}-1\right)\left(x^{2}+1\right)$$
Consider $x^{2}-1$. Rewrite $x^{2}-1$ as $x^{2}-1^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x-1\right)\left(x+1\right)$$
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: $x^{2}+1,x^{4}+2$.
