If kept in the refrigerator in an airtight container, it should last for about 1 week. If you decide to freeze your cake it can last in the freezer for 2-3 months and still thaw pretty okay. You can find helpful tips for freezing cake here: How to Freeze Cake. If you decide to add the carrots for decoration I would not save them decorated with those as the carrots may sweat into the frosting.