The [i]Relics of the Golden Age[/i] exotic quest in Destiny 2 require, among other things, that the player acquires 5 legendary engrams and 1 exotic engram. I've had the quest for a while now, and noticed two things: The first is that the legendary engram requirement is fulfilled by drops that open immediately upon pickup (which all but prime and umbral engrams do at this point); the second thing noticed is that drops labeled as "Exotic Engram", such as the rank 25 [i]Seaon of Arrivals[/i] exotic engram reward and a random exotic engram from the mail, do not fulfill the "Exotic Engrams picked up" part of the quest.
The quest does not instruct the player on how to acquire an exotic engram that doesn't auto-decrypt, and from what I've found online so far non-prime engrams that must be taken to a cryptarch have been mostly done away with. As such I am fairly confident that the quest itself is simply bugged, as it does not register the pickup of exotic engrams.
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You need to pick up an exotic dropped from a boss or ultra.