I think the most important thing which will help us to eliminate the answer choices is Subject Verb Pair
Looking at the statment 'Europeans' is Plural therefore 'have' is the correct verb and 'eating quail' is Singular therefore makes is also the correct verb, in my opinion A seems correct lets look at the other choices to see if we can find some error
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Europeans have long known that eating quail sometimes makes the eater ill, but only recently has it been established that the illness is caused by a toxin present in the quail’s body only under certain conditions.
(A) Europeans have long known that eating quail sometimes makes ->
seems Ok to me as per above evaluation(B) Europeans have long known quail eating is sometimes
able to make -> what is the need for 'able to' here??
(C)
Eating quail has long been known to Europeans to sometimes
make -> Subject Verb error as discussed above
(D) It has long been known to Europeans that quail eating
will sometimes make -> incorrect use of will, as 'sometimes make' alone tells that there is a possibility of illness
(E) It has long been known to Europeans that quail,
when it is eaten, has sometimes made -> I don't how to explain it but it's such a bad construction it seems like it is not the eating but quail itself causes the illness