\(x^8+1+x\)
\(=x^8+1+x-x^2+x^2\)
\(=\left(x^8-x^2\right)+\left(x^2+x+1\right)\)
\(=\left[\left(x^4\right)^2-x^2\right]+\left(x^2+x+1\right)\)
\(=\left(x^4-x\right)\left(x^4+x\right)+\left(x^2+x+1\right)\)
\(=x\left(x^3-1\right).x\left(x^3+1\right)+\left(x^2+x+1\right)\)
\(=x^2\left(x-1\right)\left(x^2+x+1\right)+\left(x^3+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^2\left(x-1\right)\left(x^3+1\right)+1\right)\)
\(=\left(x^2+x+1\right)\left(\left(x^3-x^2\right)\left(x^3+1\right)+1\right)\)
\(=\left(x^2+x+1\right)\left(x^6-x^5+x^3-x^2+1\right)\)
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