<s>[INST] How to compute $\frac{t^2}{t+1}$ to the form $\frac{1}{t+1} +t -1$ One of my attempts would look like below.
$\frac{t^2}{t+1}$ = $\frac{t \times t+1-1}{t+1}$ = 1+ $\frac{t-1}{t+1}$ = $\frac{t-1+1-1}{t+1} + 1$ = $\frac{-2}{t+1} +2$
Also, I put into t an arbitrary number in both equations and have gotten the same answer. The first number was 18785 to which both contained the answer $\frac{352876225}{18786}$. 
 [/INST] Several responses have you make the following factorization:
$$\frac{t^2}{t+1} = \frac{(t+1)(t-1) + 1}{t+1}$$
But in general, you can take a slightly more iterative approach:
$$\frac{t^2}{t+1} = \frac{t^2 + t - t}{t+1} = \frac{t\cdot (t+1)}{t+1} - \frac{t}{t+1} = t - \big(\frac{t+1-1}{t+1}\big) = t - 1 + \frac1{t+1}$$
You just need to consider which terms are required to turn the largest term in the numerator into a multiple of the denominator.
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