The way in which the mighty blue whale and the other baleen whales—the finback, gray, humpback, and right whales—
eats was discovered by careful observation by biologists.
A. eats was discovered by
B. eat was discovered through
C. eats were discovered by means of
D. eat were discovered by
E. eat was discovered resulting from
This is a complex sentence in which the sub -clause is enveloped within the main clause.
The main clause: The way was discovered by careful observation by biologists. The subject 'the way' is singular and therefore the verb should be the singular 'was'.
The sub-clause: in which the mighty blue whale and the other baleen whales——eat--- 'in which' is the sub -conjunction and the subject is the plural 'the mighty blue whale and the other baleen whales'. In tandem the verb should be the plural eat.
Such a combination of 'was and eat' is available in B and E, of which E suffers from bad modification, because there seems to be no noun for the participle without comma to modify. B is the eventual choice.