Preheat the oven to 450 degrees F (230 degrees C). Grease a large nonstick baking dish.
Combine potatoes, bell peppers, onion, basil, and garlic in a large mixing bowl. Lightly coat with olive oil. Sprinkle salt, black pepper, and paprika on top and toss to coat. Place in a single layer onto the prepared baking dish.
Bake in the preheated oven until potatoes are golden brown and soft, 45 minutes to 1 hour. Remove from the oven and mix in Cheddar cheese. Return to the oven and bake for an additional 5 minutes. Serve.