With her archaeologist husband, travels inspired several mystery novel

For the original poster, what's the "fundamental belief" to which you referred?

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!

With her archaeologist husband, travels inspired several mystery novels by Agatha Christie; travelers to Egypt can still stay at the Old Cataract Hotel, the model for the hotel in one of Christie's most famous books.
a. With her archaeologist husband, travels inspired several mystery novels by Agatha Christie
b. Agatha Christie used her travels with her archaeologist husband to inspire several mystery novels
c. Because her husband was an archaeologist, Agatha Christie was able to use their travels as inspiration for several of her mystery novels
d. Together with her archaeologist husband, Agatha Christie was inspired to incorporate their travel into several of her mystery novels
e. Agatha Christie's travels with her archaeologist husband served as inspiration for several of her mystery novels

OA after discussion

With her archaeologist husband, travels inspired several mystery novels by Agatha Christie; travelers to Egypt can still stay at the Old Cataract Hotel, the model for the hotel in one of Christie's most famous books.

If we delve into the meaning of each option..this one is pretty obvious
The intended meaning is X's Travel with her husband inspired several of her mystery novels

a) With her archaeologist husband, travels inspired several mystery novels by Agatha Christie-Husband and travels inspired her - Nope..travels with her husband inspired her
b) Agatha Christie used her travels with her archaeologist husband to inspire several mystery novels She did not use her travels
c) Because her husband was an archaeologist, Agatha Christie was able to use their travels as inspiration for several of her mystery novels Not that because her husband was an arch that she was able to do this
d) Together with her archaeologist husband, Agatha Christie was inspired to incorporate their travel into several of her mystery novels Husband and agatha both were inspired..this is incorrect..it was ply Agatha who was inspired
e) Agatha Christie's travels with her archaeologist husband served as inspiration for several of her mystery novels