Thelonious Monk...

IMO E

The answer is D, right?

Thelonious Monk, who was a jazz pianist and composer, produced a body of work both rooted in the stride-piano tradition of Willie Smith and Duke Ellington, yet in many ways he stood apart from the mainstream jazz repertory.

A)Same - both is redundant here, hence wrong option
B) Thelonious Monk, the jazz pianist and composer, produced a body of work that was rooted in both - both is redundant here, hence wrong option
C) Jazz pianist and composer Theloniosus Monk, produced a body of work that was rooted - produced....Ellington - it can be written more clearly as part of the main sentence
D) Jazz pianist and composer Thelonious Monk produced a body of work that was rooted - correct answer
E) Jazz pianist and composer Thelonious Monk produced a body of work rooted both - both is redundant here, hence wrong option

Nice explanation, Sonia!

One big strategic point here - if you see the word "both", it begs for parallel structure. So you can scan for the "and" and determine what you're working with on either side. Here, we only have "Duke Ellington" after "and", so we can't have anything between "both" and "Willie Smith" or that parallel structure is broken. And since the non-underlined portion has "...in the stride-piano tradition of...", "both" can't work in any of these options. In order to use both, we'd have to say:

...in the stride-piano tradition of BOTH Smith and Ellington

or

...in both the stride-piano tradition of Smith and the ___________ style of Ellington

or

...both in the stride-piano tradition of Smith and in the __________style of Ellington

"Both" is a great signal for you that you need to find parallel structure between two things, and that "both" is the starting point. You can almost envision it as the pull-tab portion of a zipper - wherever "both" exists, the two sections that diverge must be equal or the sentence is wrong (kind of like if you ever notice that one length of the zipper is significantly longer than the other...look down and you'll find that the part you already zipped is screwed up).