Typically you would try to get all of the variables on one side, and all of the numbers on the other.
So in this instance, you’d start by moving them around to get things together:
x+2 = x-2
x+2-x = -2
x-x = -2-2
But then you simplify, and cancel out any variables that need to be cancelled. In this case we see “x-x” so that cancels out to 0. And we see -2-2 which simplifies into -4. So the end result is:
0=-4
Which is obviously a nonsense answer. In the original post, homeslice did the first step wrong, moved everything over to the left incorrectly, (inadvertently setting the whole equation equal to 0) and the whole thing was downhill from there; Since the first step of their solution was wrong, everything behind it was also wrong.