As the two things mentioned prior to the name of Franklin are the only reasons being predicted to be related to Guillain-Barre syndrome, there should be 'and' in between them and not a comma. Hence 'D' is out.
'has' is not the correct tense for representing the adulthood. Hence 'B' is out.
'having not experienced' is not proper and hence 'C' is out.
Out of the remaining options 'A' and 'E',as 'until adulthood' is used, past perfect is more suitable and hence option 'A' is more appropriate.

I think people are getting confused here about the basic meaning of the sentence:

Eliminating B and D on the basis of tense. Has not/ had been/had been

We are left with A, C, E

Check A first:
if one can't understand the meaning, test the timeline:
Past perfect----->Past------>Present
He had not experienced any symptoms till adulthood (From birth till adulthood)----->He was facing facial paralysis (which according to passage suggests is a symptom of GBS and not complications of Polio)----->Diagonis may be wrong (suggestion in present)

C's timeline
Early on he experienced problems and having not experienced...hhmmmmmmm....
----> He had facial paralysis and also did not have any symptoms (Overlap of timeline....because past and present perfect continuous suggests continuation till present) -- > Total confusion here!

E's timeline
He did not face symptoms but had facial paralysis
Its like the best contradiction ever!
How can a person not face any symptom but have facial paralysis?
E negated.

Left with A
So OA is A

Hope this helps!

Because he had not experienced any symptoms until adulthood and was affected by facial paralysis early on in the course of his illness, Franklin Delano Roosevelt, whose death was originally attributed to complications arising from polio, may actually have suffered from symptoms related to Guillain-Barre syndrome.

(A) Because he had not experienced any symptoms until adulthood and was affected by facial paralysis early on in the course of his illness Correct Answer
(B) Since he has experienced his symptoms in adulthood and has been affected by facial paralysis early on in the course of his illness - this structure also seems to change the meaning of the sentence so I would cross this off
(C) Early on in the course of his illness he experienced facial paralysis and having not experienced any symptoms until adulthood - notice the "and" - there is no parallel structure before and after "and"
(D) Because he had not experienced any symptoms until adulthood, he had been affected by facial paralysis early on in the course of his illness - Based on the below use of "had" this is also incorrect. "Had" must be used in conjunction with the simple past (e.g., was, did)
(E) Since he did not experience any symptoms until adulthood and had been affected by facial paralysis early on in the course of his illnessRecall the use of "had" - it is a past perfect verb, meaning it is "the past of the past". It does not make for him to have been affected by the facial paralysis prior to growing up. The incorrect use of "had" here switches the sequence of events.