Not taking them out starts the job faster but then after your install there will be quarter round regret because of point 1.
Remove quarter round before carpet.
Go along the quarter round until there is a natural break in the molding such as a corner or mitered edge.
Some put the base molding down first then carpet others put the carpet down then base molding and sometimes base shoe but i find it a bit unsightly.
Instead simply pick up some quarter round trim which generally measures about 3 4 inch high and can be found at any hardware or home improvement store.
Insert the blade of a pry bar behind the quarter round at the nailing points visible between the quarter round and baseboard.
Touch up paint and caulk where needed.
Take the baseboards out freshen them up while you are at it install floor pop baseboards back.
For the quarter round experts.
Wiggle the putty knife back and forth along this seam until the quarter round pops loose from the wall.
The contractor placed the flooring but there are a lot of dips.
Cut it to length using a miter saw to angle the ends then use finish nails to attach it to the baseboard making sure not to nail into the floor itself.
Pry outward gently moving from one nailing point to the next.
Do this all around the room.
It depends on the pile height of your new carpet and how high your baseboards are.
Carpet can stain and wear out over time so plan to replace it every five to ten years especially if it is installed in high traffic areas like living rooms.
Quarter round is inappropriate base shoe is the proper stuff but not with carpet.
To aid in re installing the quarter round number each piece in order as you remove it around the room.
Often a quarter round piece of molding can be run along the bottom of the baseboard to finish it off.