Divide $$$x^{6}$$$ by $$$\left(x^{2} + 1\right)^{2}$$$

Rewrite the divisor: $$$\left(x^{2} + 1\right)^{2} = x^{4} + 2 x^{2} + 1$$$.

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

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

Step 1

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

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

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

Subtract the dividend from the obtained result: $$$\left(x^{6}\right) - \left(x^{6}+2 x^{4}+x^{2}\right) = - 2 x^{4}- x^{2}$$$.

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

Step 2

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

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

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

Subtract the remainder from the obtained result: $$$\left(- 2 x^{4}- x^{2}\right) - \left(- 2 x^{4}- 4 x^{2}-2\right) = 3 x^{2}+2$$$.

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

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

The resulting table is shown once more:

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

Therefore, $$$\frac{x^{6}}{\left(x^{2} + 1\right)^{2}} = \left(x^{2} - 2\right) + \frac{3 x^{2} + 2}{\left(x^{2} + 1\right)^{2}}$$$.