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Remove the stalks from the bulb. Place the bulbs and stalks into a clean bag to store them for 10 days. Put the fennel in a glass container and cover it with water to store it for 5 days. Place the packaged fennel in the refrigerator.
Using a sharp knife, carefully cut the stalk about 1 inch (2.5 cm) above the bulb. Be careful not to place too much pressure on the bulb, as they can bruise easily and subsequently rot. Discard any stalks or leafy sections that are brown or wilted. Do not wash the stalks or bulbs after cutting and/or prior to placing in storage containers. You can use either plastic or paper bags. There is no need to store the stalks and bulbs in separate bags, but you can if you choose. Separate bags would be a good idea if you know they will have separate uses.  Roll the bag up carefully. Start at the bottom where the herbs are, and while gently holding them in place, roll the bag up towards the top opening. Take care not to damage the bulbs by applying too much pressure. Do not seal the bag shut. Some people feel that plastic bags or containers are unsanitary, and would rather use glass. This is fine, just be sure to use cold water and seal the container tightly. Avoid storing the fennel in the coldest areas of your fridge, like in the back, because it can cause the fennel to partially freeze which will ruin the texture and the flavor. Instead, place the fennel near the front of the vegetable crisper or shelf until you are ready to use. Fennel stored in a bag will last 7-10 days in the refrigerator. Stored in a glass container, it will only last 3-5 days.