Answer
$\text{The factored form of the given expression is }$$\displaystyle \left( x^{4} +x^{2} +1\right)\left( x^{4} -x^{2} +1\right).$
Work Step by Step
$\displaystyle \begin{array}{{>{\displaystyle}l}}
\begin{array}{ c l l }
x^{8} +x^{4} +1 & =x^{8} +2x^{4} +1-x^{4} & \mathrm{Add} \ x^{4} \text{ to the second term and subtract } x^4. \\
& & \\
& =\left( x^{8} +2x^{4} +1\right) -x^{4} & \begin{array}{{>{\displaystyle}l}}
\mathrm{Apply\ associative\ property}\\
( a+b) +c=a+( b+c)
\end{array}\\
& & \\
& =\left( x^{4} +1\right)^{2} -\left( x^{2}\right)^{2} & \begin{array}{{>{\displaystyle}l}}
\mathrm{Apply\ special\ products\ property}\\
\left( a^{2} +2ab+b^{2}\right) =( a+b)^{2}
\end{array}\\
& & \\
& =\left( x^{4} +1+x^{2}\right)\left( x^{4} +1-x^{2}\right) & \begin{array}{{>{\displaystyle}l}}
\mathrm{Apply\ the\ difference\ of\ squares\ property}\\
a^{2} +b^{2} =( a+b)( a-b)
\end{array}\\
& & \\
& =\left( x^{4} +x^{2} +1\right)\left( x^{4} -x^{2} +1\right) & \begin{array}{{>{\displaystyle}l}}
\mathrm{Apply\ the\ commuative\ property}\\
a+b=b+a
\end{array}
\end{array}
\end{array}$
