To factor the expression, solve the equation where it equals to $0$.
$$a^{4}+a^{2}-2=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-2$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.
$$±2,±1$$
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
$$a=1$$
By Factor theorem, $a-k$ is a factor of the polynomial for each root $k$. Divide $a^{4}+a^{2}-2$ by $a-1$ to get $a^{3}+a^{2}+2a+2$. To factor the result, solve the equation where it equals to $0$.
$$a^{3}+a^{2}+2a+2=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $2$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.
$$±2,±1$$
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
$$a=-1$$
By Factor theorem, $a-k$ is a factor of the polynomial for each root $k$. Divide $a^{3}+a^{2}+2a+2$ by $a+1$ to get $a^{2}+2$. To factor the result, solve the equation where it equals to $0$.
$$a^{2}+2=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $1$ for $a$, $0$ for $b$, and $2$ for $c$ in the quadratic formula.
$$a=\frac{0±\sqrt{0^{2}-4\times 1\times 2}}{2}$$
Do the calculations.
$$a=\frac{0±\sqrt{-8}}{2}$$
Polynomial $a^{2}+2$ is not factored since it does not have any rational roots.
$$a^{2}+2$$
Rewrite the factored expression using the obtained roots.
