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The second sentence mentions publication in The Adventures of Tom Bombadil, but the first sentence only says "first published in 1933." Consider adding that it was first published in The Oxford Magazine.
Done.
"in revised and extended form" -> "in a revised and extended form"
In Britspeak it's right as it is, if that's ok with you.
"as described in The Lord of the Rings.": most readers are going to know that Tolkien wrote LotR, but some readers may not, so the book should be introduced as "Tolkien's 1954–55 novel", or some variation.
Done.
"Christina Scull & Wayne G. Hammond": these two need introductions
Done.
"quartet": can this be piped to quatrain? It's not a term I'm familiar with.
Done. The terms are equivalent so let's go with quatrain.
I was mentioning the use in the body: "The second and fourth line in every quartet rhyme..." Since you were okay with changing it in the infobox, I've gone ahead and changed it in the body as well.
"This was so difficult that he" -> He found this so difficult that he"
Image looks good and is properly tagged. Caption can be made active as 'Gilbert and Sullivan's "Major-General's Song" inspired the poem'.
We could, but the emphasis here is on "The poem".
Yes, I agree. Better to keep it as is.
"The poem was set to music by the composer and entertainer Donald Swann" can be made active as "The composer and entertainer Donald Swann set the poem to music."
pass: there was one other use of quartet in the body that I went and changed to quatrain (mentioned above) but that was a small quibble. This is a wonderful page! Happy to give it a pass.