The main problem with the sentence as written is the modifier at the end of the sentence. What has been ignored by the mainstream press is "a few private corporations," but starting the modifier with "having" means it should refer all the way to "they"--modern methods of voter registration. The mistake is repeated in (E), so we're left with the middle three choices.

While the wording is different, the same mistake occurs in (C). "and have" turns "been ignored..." into the second part of a list, the first part is "placed ... trust." Again, the grammar indicates that the modern methods have been ignored by the press. That leaves (B) and (D).

Choice (D) is an incomplete sentence. "placing unprecedented trust..." is a modifier of something that should come after it. However, there is no subject that corresponds with it. That leaves only (B), a complete sentence that correctly indicates that the private corporations are what has been ignored by the press. (B) is correct.