The correct option is C
neither does Ken
In the given sentence, 'and' acts as a conjunction of affirmation. It indicates that the second clause supports or acts in favour of the first. As we can see, the sentence is about Helen and Ken eating potatoes. Since the first clause says that Helen does not eat potatoes, the second clause should support it by saying that Ken does not eat potatoes either.
Option A is incorrect. In 'so doesn't Ken', 'so' indicates a reason. Here, the answer must simply indicate that Ken also does not eat potatoes, there is no reason indicated. Thus, so does not fit into the blank.
Option B is incorrect. 'neither doesn't Ken' indicates that Ken eats potatoes and also, Helen eats them too. Thus, it does not convey the correct meaning.
Option C is correct. 'neither does Ken' indicates that Ken also does not eat potatoes. Thus, it conveys the correct meaning and is the correct answer.
Option D is incorrect. 'neither Ken does' is a grammatically incorrect phrase. The main verb of the clause is placed after the object. The main verb should come before the object and so, it is grammatically incorrect.