When potatoes go bad they will begin to get soft and wither which is a sign they are either bad or going bad.
Other signs that potatoes are going bad or are bad are discoloration and growths through the skin of the potato and also bad smells.
When potatoes are going bad they will start to leak out the water in a sour state and the potatoes and that leaking water form the potatoes will smell bad.
Potatoes usually last a week or 2 weeks at most after you buy them from the store when you leave them out at room temperature.
If you want to make the potatoes last longer than a week or two weeks then you need to keep the potatoes stored in a cool dark place.
Or you can place the potatoes in the refrigerator or freezer to make them last longer.
I've kept potatoes frozen in the freezer for up to a year before and they've been fine to make fries etc with or to make mashed potatoes with.
Although the texture of the frozen potatoes will change and they won't taste as good when frozen but it makes the potatoes last longer when frozen.