Divide $$$u^{4}$$$ by $$$u^{2} + 1$$$

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

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

Step 1

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

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

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

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

$$\begin{array}{r|rrrrr:c}&{\color{Blue}u^{2}}&&&&&\\\hline\\{\color{Magenta}u^{2}}+1&{\color{Blue}u^{4}}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{Blue}u^{4}}}{{\color{Magenta}u^{2}}} = {\color{Blue}u^{2}}\\&-\phantom{u^{4}}&&&&&\\&u^{4}&+0 u^{3}&+u^{2}&&&{\color{Blue}u^{2}} \left(u^{2}+1\right) = u^{4}+u^{2}\\\hline\\&&&- u^{2}&+0 u&+0&\end{array}$$

Step 2

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

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

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

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

$$\begin{array}{r|rrrrr:c}&u^{2}&{\color{Purple}-1}&&&&\\\hline\\{\color{Magenta}u^{2}}+1&u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\\&-\phantom{u^{4}}&&&&&\\&u^{4}&+0 u^{3}&+u^{2}&&&\\\hline\\&&&{\color{Purple}- u^{2}}&+0 u&+0&\frac{{\color{Purple}- u^{2}}}{{\color{Magenta}u^{2}}} = {\color{Purple}-1}\\&&&-\phantom{- u^{2}}&&&\\&&&- u^{2}&+0 u&-1&{\color{Purple}-1} \left(u^{2}+1\right) = - u^{2}-1\\\hline\\&&&&&1&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrrrr:c}&{\color{Blue}u^{2}}&{\color{Purple}-1}&&&&\text{Hints}\\\hline\\{\color{Magenta}u^{2}}+1&{\color{Blue}u^{4}}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{Blue}u^{4}}}{{\color{Magenta}u^{2}}} = {\color{Blue}u^{2}}\\&-\phantom{u^{4}}&&&&&\\&u^{4}&+0 u^{3}&+u^{2}&&&{\color{Blue}u^{2}} \left(u^{2}+1\right) = u^{4}+u^{2}\\\hline\\&&&{\color{Purple}- u^{2}}&+0 u&+0&\frac{{\color{Purple}- u^{2}}}{{\color{Magenta}u^{2}}} = {\color{Purple}-1}\\&&&-\phantom{- u^{2}}&&&\\&&&- u^{2}&+0 u&-1&{\color{Purple}-1} \left(u^{2}+1\right) = - u^{2}-1\\\hline\\&&&&&1&\end{array}$$

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