Divide $$$x^{3} - 2 x^{2}$$$ by $$$x^{2} + 1$$$

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

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

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{x^{3}}{x^{2}} = x$$$.

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

Multiply it by the divisor: $$$x \left(x^{2}+1\right) = x^{3}+x$$$.

Subtract the dividend from the obtained result: $$$\left(x^{3}- 2 x^{2}\right) - \left(x^{3}+x\right) = - 2 x^{2}- x$$$.

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

Step 2

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

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

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

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

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

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

The resulting table is shown once more:

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

Therefore, $$$\frac{x^{3} - 2 x^{2}}{x^{2} + 1} = \left(x - 2\right) + \frac{2 - x}{x^{2} + 1}$$$.