Divide $$$x^{4}$$$ by $$$x^{2} - 1$$$

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

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

Step 1

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

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

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

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

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

Step 2

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

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

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

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

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

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

The resulting table is shown once more:

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

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