You take $\displaystyle 1+s^2~|\overline{~~~~s^2}$
We see that $\displaystyle s^2$ goes into $\displaystyle s^2$ 1 time.
Thus, we should get
$\displaystyle .~~~~~~~~~~~~~~~~~~{\color{blue}1}$
$\displaystyle {\color{magenta}1+s^2}~|\overline{~~~~~~~~~s^2}$
$\displaystyle .~~~~~~~~~\underline{-(1+s^2)}$
$\displaystyle .~~~~~~~~~~~~~~~~~{\color{red}-1}$
So our quotient will be $\displaystyle {\color{blue}1}+\frac{{\color{red}-1}}{{\color{magenta}1+s^2}}=1-\frac{1}{1+s^2}$, where $\displaystyle -\frac{1}{1+s^2}$ is the remainder.
Does this make sense?
--Chris
This is done by polynomial long division. Check the link here to see how it is done: Polynomial Long Division
If you want to do it quickly without polynomial division:
$\displaystyle \frac{x^2}{x^2+1} $
Add one/subtract one: $\displaystyle \frac{(x^2 + 1) - 1}{x^2+1}$
Split: $\displaystyle \frac{x^2+1}{x^2+1} - \frac{1}{x^2+1}$
Cancel: $\displaystyle 1 - \frac{1}{x^2+1} $