Answer: Japanese painted fern (Athyrium niponicum) thrives in shaded moist garden conditions with loose, well-draining soil. It is considered a deciduous fern, meaning it naturally dies back in the fall when weather turns, but its root system (called rhizomes) will survive and send up new fronds next spring. It would be difficult to recreate these conditions in a dorm room, so it might be better to return it home and let it die back naturally outdoors. If protected with a layer of mulch, it should survive, although they do better in the ground than in pots. You're correct; those are spores.
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