polynomials – Restricting the domain of an Inverse Function

I stumbled upon a question in PSet 1 of MIT’s 18.01sc:

Find the inverse to this function: x^2 + 2x (Restrict the domain if necessary). The answer is as follows:

Restrict domain to x ≤ −1, so when it’s flipped about the diagonal y = x, you’ll still get the graph of a function. Solving for x, we get x = √(y + 1) − 1, so the inverse function is y = (√x + 1) − 1

I am kinda confused about 2 things (which I think are related):

Why did we restrict the domain to x>1 rather than x<1, and when exactly during our work did we come to the conclusion that we need to restrict it (as I’ve failed to reach the same kind of reasoning by trying to restrict the domain after already having the equation in it’s new form).

How did we come to only consider the positive root of (x+1) in the final equation? (I do realize that we cannot have the 2 roots altogether so that it doesn’t violate a function’s mapping property, yet I do not understand why did we chose the positive one, and also when did we do that).

Please pardon the repetition in case the 2 questions are basically about the same thing, I am very puzzled though and cannot make the connection, thereby would greatly appreciate if anyone could go through the middle steps..

Thanks!!