List all the elements of $A = \{ n \in \mathbb{Z} \mid \frac{n^2-n+2}{2n+1} \in \mathbb{Z}\}$ I was given the following set $A = \{ n \in \mathbb{Z} \mid \frac{n^2-n+2}{2n+1} \in \mathbb{Z}\}$
I have to list all the elements of $A$.
I started using the Euclidean division:
$$n^2-n+2=(2n+1)(\frac{1}{2}n-\frac{3}{4}) + \frac{11}{4}$$
In order to eliminate the fractions, I multiplied the whole expression with 4:
$$4(n^2-n+2)=(2n+1)(2n-3) + 11$$
I´m stuck here, maybe I´m on the wrong way. Could anyone give me some hints how can I approach this problem. Thank you in advance.

As have you have noticed,
$$
4\frac{n^2-n+2}{2n+1}= 2n-3+ \frac{11}{2n+1}
$$
Therefore, if $\dfrac{n^2-n+2}{2n+1}$ is an integer, then so is $\dfrac{11}{2n+1}$.
This leaves very few candidates for $n$.
You need to check each candidate to see whether $\dfrac{n^2-n+2}{2n+1}$ is an integer.
