<s>[INST] How does $\frac{t^2}{t+1}$ equal $t-1+\frac{1}{t+1}$? I do the long division:
1: t+1 goes into $t^2$ t times
2: Subtract $t^2$ + 1 from $t^2$ and get -1
3: Answer:  t - $\frac{1}{t+1}$
Am I missing something here?
 [/INST] Polynomial division will also give you the answer quickly:
$$\begin{array}{cccccccc}
     &   &    & t   & &  -1  &  & \\
(t+1)& | &    & t^2 & &  +0t &  &+0  \\
     &   & -( & t^2 & &  +t  &) &     \\
     &   &    &     & & -t   &  & +0 \\
     &   &    &     &-(&-t   & -1 & )\\
     &   &    &     &  &     & +1\\
\end{array}$$
$$\textrm{Result : } \boxed{t- 1 + \dfrac{1}{t+ 1}}.$$
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