Pretty much, if they give you a detail on quant, it's because that detail is important in some way. So, first, write EVERYthing down. Everything.
Go through a bunch of problems and just write stuff down. Develop a
consistent shorthand for yourself. When they tell me x is a positive integer, I always write x = pos int. When they tell me x is positive, I write x=+. You don't have to write things exactly the way I do, but you do need to do the same thing for the same piece of information every time.
The act of writing down "x=+" will automatically put that piece of info a little higher in the consciousness of your brain. You will also need to keep repeating to yourself what I told you up above: if they bother to give you a constraint, there's a reason. Until you've figured out what that reason is, you can't forget about that constraint.
Some constraints are more important than others. For instance, sometimes they tell me x does not equal zero, and then I notice that they use x on the bottom of a fraction. Okay, so they told me x isn't equal to zero because you can't have zero on the bottom of a fraction. Now I can forget about that constraint, and I didn't actually have to do anything with it. On other constraints, I will very much have to use the constraint actively.
So two mantras:
1) if they give me a constraint, it's important for the functioning of the problem; I can't ignore it till I figure out WHY they gave it to me
2) I write EVERYthing down, including all the little details / constraints (though I use shorthand - I don't want to transcribe the entire problem!)
Next, take a look at this article about how to minimize the mistakes that you make. (Note: you're always going to make some careless mistakes - don't expect to be perfect. But it is a good goal to try to minimize them as much as possible. :)
http://www.manhattangmat.com/articles/error-log.cfm