You must have superhero carrots, without other sort of intervention I reckon 4-5 days can be pushing it! Mould, going soggy etc. is the usual.
I use the magic green bags which absorb some gas or another and keeps them fresh for longer, some fridges also now have this technology - I find 4-6 weeks is ok.
I think also if you have somewhere suitable to store them in cool, dark and dry conditions they can last a while - a lot of modern homes lack this but if you live in an older place with a larder or root store (or old coal shed that rats cannot get into) that would be ideal.
But yeah there isn't a use by date on carrots - if they aren't mouldy or disintegrating and smell and taste ok, then they are fine to use.