Why don't we include $\pm\infty$ in $\mathbb R$?

By defining, for example, $+ \infty + n:= + \infty\ \forall n \in\mathbb{R}$ , we'd lose the group structure and this is no good for lots of purposes.

A large reason why we don't include $\infty$ is because we can't really do arithmetic with it. $\mathbb{R}$ is a field, meaning that is satisfies a list of axioms that give it a certain structure. I suggest you look up these axioms if you're not familiar with them and see just how many of them start to fail if you try to include an $\infty$ in $\mathbb{R}$.

Yes, including $\infty$ may give you a few extra solutions to some problems, but it won't solve every problem ($x^2=-2$ for example) and in some cases, it will introduce answers you probably don't want (would $x=x+2$ now have the solution $x=\infty$?). All of this isn't really worth all of the issues you get for including $\infty$ in $\mathbb{R}$.

There is no reason we can't add $\pm\infty$ to our set. Sometimes it is actually convenient to do so: in measure theory, it's common to work with "the extended non-negative reals" $\mathbb R_{\geq 0}\cup\{+\infty\}$ because it simplifies some common statements.

For doing arithmetic, it is very inconvenient however. In our standard $\mathbb R$, whenever we divide by something, we always have to make sure the denominator is not $0$, since dividing by $0$ is not allowed. This is an annoying artifact we have to live with as long as we want to divide things. Likewise, if we added $\pm\infty$ to our set, there is no good definition for what $\infty-\infty$ is, so whenever we subtract two variables, we have to make sure our numbers are not $\infty$. If we add two numbers $x+y$, and we happen to have $x=\infty$ and $y=-\infty$, we run into the same problem, so we can't even add two numbers without first checking they're not infinite! This is tedious, so we just decide not to include $\pm\infty$ in our set.