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

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

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

Step 1

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

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

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

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

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

Step 2

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

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

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

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

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

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

The resulting table is shown once more:

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

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