Fenmarch was a region of eastern Rohan that stretched from the Firien Wood along the Mering Stream, and therefore marked part of Rohan's eastern border with Gondor.
The fen element of the name means 'marsh' and is a reference to the wetlands along the Stream. March is an old word for 'borderland' - a reference to the border with Anórien - and has the same derivation as Mark, a name for the whole country of Rohan.
Note that Tolkien writes that on reflection he would have preferred the form Fenmark, to the extent that he recommended this alternative version for translated versions of the book.
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