Divide $$$u^{2}$$$ by $$$1 - u$$$

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

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

Step 1

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

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

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

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

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

Step 2

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

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

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

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

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

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

The resulting table is shown once more:

$$\begin{array}{r|rrr:c}&{\color{DarkCyan}- u}&{\color{Chartreuse}-1}&&\text{Hints}\\\hline\\{\color{Magenta}- u}+1&{\color{DarkCyan}u^{2}}&+0 u&+0&\frac{{\color{DarkCyan}u^{2}}}{{\color{Magenta}- u}} = {\color{DarkCyan}- u}\\&-\phantom{u^{2}}&&&\\&u^{2}&- u&&{\color{DarkCyan}- u} \left(- u+1\right) = u^{2}- u\\\hline\\&&{\color{Chartreuse}u}&+0&\frac{{\color{Chartreuse}u}}{{\color{Magenta}- u}} = {\color{Chartreuse}-1}\\&&-\phantom{u}&&\\&&u&-1&{\color{Chartreuse}-1} \left(- u+1\right) = u-1\\\hline\\&&&1&\end{array}$$

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