It's typically most efficient to look for splits, but the fact that there is
not an obvious split at the beginning of the sentence is a clue that we may be dealing with a MODIFIER issue.
(A) The prepositional phrase
"with her archaeologist husband" incorrectly modifies "TRAVELS INSPIRED" (and implies that the HUSBAND and the TRAVELS were the source of inspiration, rather than the travels that she took with her husband).
Eliminate.
(B) Unlike in choice A, there is no dangling modifier at the beginning here, but meaning-wise this choice is a little wonky. She USED her travels to inspire?
Eliminate.(C) The initial modifier is correct here, but notice that little pronoun "
their" midway through the sentence. "Agatha Christie and her husband" has not appeared as a compound subject (even though we know that's what the sentence writer probably *means*, we can only treat what is
actually present).
Eliminate.
(D) Slightly sneakier version of choice C here. Even though "together with" lets us know that the sentence writer probably means "Agatha and her husband," the only word that would give us a compound subject is "AND." Without that conjunction, the pronoun "their" is still incorrect.
Eliminate.(E) No dangling or misplaced modifiers here. The travels themselves have become the subject, and the possessive pronoun "her" correctly refers back to the possessive antecedent "Agatha Christie's." This is our answer!