In the TNG episode "Where no one has gone before", the Enterprise-D went to the Triangulum galaxy.
Do you think the Federation just forgot about that? Or would they try to figure out how to replicate the experiment?
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I'm sure they would have wanted to, but it's made abundantly clear at the end of the episode that it's not the Federation's warp enhancements that made the jump possible, it was the Traveller's abilities
PICARD: (to Kosinski) What did you do?
RIKER: It wasn't him. It never was. It was his assistant.
PICARD: What are you talking about?
RIKER: Kosinski wasn't the one controlling the warp experiments.
KOSINSKI: It was me!
RIKER: The equations he punched in were nonsense, just as we thought.
At the end of the episode the Traveller disappears and doesn't return until a later episode. In his absence they only have sensor readings (that aren't especially helpful) and Kosinski's worthless equations.
RIKER: The Traveller's gone, sir.
RIKER: He's phased completely out of existence. At least, out of our existence.
Do you think the Federation just forgot about that?
Why would you think that? The fact that it wasn't discussed explicitly in TNG doesn't mean it was ignored or forgotten. Just not featured in an episode.
Or would they try to figure out how to replicate the experiment?
Maybe. Those brainiacs at the Daystrom institute's Mars college (where Lea Brahms did her Ph.D. research) are looking into exotic propulsion theories all the time.