Divide $$$x^{6} - 1$$$ by $$$x^{2} + 1$$$

Write the problem in the special format (missed terms are written with zero coefficients):

$$$\begin{array}{r|r}\hline\\x^{2}+1&x^{6}+0 x^{5}+0 x^{4}+0 x^{3}+0 x^{2}+0 x-1\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{x^{6}}{x^{2}} = x^{4}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$x^{4} \left(x^{2}+1\right) = x^{6}+x^{4}$$$.

Subtract the dividend from the obtained result: $$$\left(x^{6}-1\right) - \left(x^{6}+x^{4}\right) = - x^{4}-1$$$.

$$\begin{array}{r|rrrrrrr:c}&{\color{DarkBlue}x^{4}}&&&&&&&\\\hline\\{\color{Magenta}x^{2}}+1&{\color{DarkBlue}x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&-1&\frac{{\color{DarkBlue}x^{6}}}{{\color{Magenta}x^{2}}} = {\color{DarkBlue}x^{4}}\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+x^{4}&&&&&{\color{DarkBlue}x^{4}} \left(x^{2}+1\right) = x^{6}+x^{4}\\\hline\\&&&- x^{4}&+0 x^{3}&+0 x^{2}&+0 x&-1&\end{array}$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- x^{4}}{x^{2}} = - x^{2}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$- x^{2} \left(x^{2}+1\right) = - x^{4}- x^{2}$$$.

Subtract the remainder from the obtained result: $$$\left(- x^{4}-1\right) - \left(- x^{4}- x^{2}\right) = x^{2}-1$$$.

$$\begin{array}{r|rrrrrrr:c}&x^{4}&{\color{Chartreuse}- x^{2}}&&&&&&\\\hline\\{\color{Magenta}x^{2}}+1&x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&-1&\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+x^{4}&&&&&\\\hline\\&&&{\color{Chartreuse}- x^{4}}&+0 x^{3}&+0 x^{2}&+0 x&-1&\frac{{\color{Chartreuse}- x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Chartreuse}- x^{2}}\\&&&-\phantom{- x^{4}}&&&&&\\&&&- x^{4}&+0 x^{3}&- x^{2}&&&{\color{Chartreuse}- x^{2}} \left(x^{2}+1\right) = - x^{4}- x^{2}\\\hline\\&&&&&x^{2}&+0 x&-1&\end{array}$$

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{x^{2}}{x^{2}} = 1$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$1 \left(x^{2}+1\right) = x^{2}+1$$$.

Subtract the remainder from the obtained result: $$$\left(x^{2}-1\right) - \left(x^{2}+1\right) = -2$$$.

$$\begin{array}{r|rrrrrrr:c}&x^{4}&- x^{2}&{\color{Fuchsia}+1}&&&&&\\\hline\\{\color{Magenta}x^{2}}+1&x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&-1&\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+x^{4}&&&&&\\\hline\\&&&- x^{4}&+0 x^{3}&+0 x^{2}&+0 x&-1&\\&&&-\phantom{- x^{4}}&&&&&\\&&&- x^{4}&+0 x^{3}&- x^{2}&&&\\\hline\\&&&&&{\color{Fuchsia}x^{2}}&+0 x&-1&\frac{{\color{Fuchsia}x^{2}}}{{\color{Magenta}x^{2}}} = {\color{Fuchsia}1}\\&&&&&-\phantom{x^{2}}&&&\\&&&&&x^{2}&+0 x&+1&{\color{Fuchsia}1} \left(x^{2}+1\right) = x^{2}+1\\\hline\\&&&&&&&-2&\end{array}$$

Since the degree of the remainder is less than the degree of the divisor, we are done.

The resulting table is shown once more:

$$\begin{array}{r|rrrrrrr:c}&{\color{DarkBlue}x^{4}}&{\color{Chartreuse}- x^{2}}&{\color{Fuchsia}+1}&&&&&\text{Hints}\\\hline\\{\color{Magenta}x^{2}}+1&{\color{DarkBlue}x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&-1&\frac{{\color{DarkBlue}x^{6}}}{{\color{Magenta}x^{2}}} = {\color{DarkBlue}x^{4}}\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+x^{4}&&&&&{\color{DarkBlue}x^{4}} \left(x^{2}+1\right) = x^{6}+x^{4}\\\hline\\&&&{\color{Chartreuse}- x^{4}}&+0 x^{3}&+0 x^{2}&+0 x&-1&\frac{{\color{Chartreuse}- x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Chartreuse}- x^{2}}\\&&&-\phantom{- x^{4}}&&&&&\\&&&- x^{4}&+0 x^{3}&- x^{2}&&&{\color{Chartreuse}- x^{2}} \left(x^{2}+1\right) = - x^{4}- x^{2}\\\hline\\&&&&&{\color{Fuchsia}x^{2}}&+0 x&-1&\frac{{\color{Fuchsia}x^{2}}}{{\color{Magenta}x^{2}}} = {\color{Fuchsia}1}\\&&&&&-\phantom{x^{2}}&&&\\&&&&&x^{2}&+0 x&+1&{\color{Fuchsia}1} \left(x^{2}+1\right) = x^{2}+1\\\hline\\&&&&&&&-2&\end{array}$$

Therefore, $$$\frac{x^{6} - 1}{x^{2} + 1} = \left(x^{4} - x^{2} + 1\right) + \frac{-2}{x^{2} + 1}$$$.