Final answer to the problem
$x^{7}+x^{6}+x^{5}+x^{3}-1$
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Divide $x^{10}+2x^6+x^2-1$ by $x^3-x^2+1$
$\begin{array}{l}\phantom{\phantom{;}x^{3}-x^{2}+1;}{\phantom{;}x^{7}+x^{6}+x^{5}\phantom{-;x^n}+x^{3}\phantom{-;x^n}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{;}x^{3}-x^{2}+1\overline{\smash{)}\phantom{;}x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{3}-x^{2}+1;}\underline{-x^{10}+x^{9}\phantom{-;x^n}-x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{10}+x^{9}-x^{7};}\phantom{;}x^{9}\phantom{-;x^n}-x^{7}+2x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-x^{2}+1-;x^n;}\underline{-x^{9}+x^{8}\phantom{-;x^n}-x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-x^{9}+x^{8}-x^{6}-;x^n;}\phantom{;}x^{8}-x^{7}+x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-x^{2}+1-;x^n-;x^n;}\underline{-x^{8}+x^{7}\phantom{-;x^n}-x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-x^{8}+x^{7}-x^{5}-;x^n-;x^n;}\phantom{;}x^{6}-x^{5}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-x^{2}+1-;x^n-;x^n-;x^n;}\underline{-x^{6}+x^{5}\phantom{-;x^n}-x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-x^{6}+x^{5}-x^{3}-;x^n-;x^n-;x^n;}-x^{3}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-x^{2}+1-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}x^{3}-x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{;;;;\phantom{;}x^{3}-x^{2}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\\\end{array}$
$x^{7}+x^{6}+x^{5}+x^{3}-1$
Final answer to the problem
$x^{7}+x^{6}+x^{5}+x^{3}-1$