Wash and cut the potatoes into bite-sized pieces. Leave the mushrooms whole. In a large bowl, combine the artichokes, potatoes, mushrooms, and red peppers, olive oil, salt, cayenne pepper and toss everything together until well coated. Spread the vegetables out on a large baking sheet in a single layer. Roast in the preheated oven for about 40-45 minutes, removing the mushrooms after 10 minutes of baking. Stir occasionally to ensure even roasting.