A recent poll indicates that many people in the United States hold a combination of conservative and liberal political views; i.e.,
they denounce big government, saying government is doing too much and has become too powerful, while at the same time supporting many specific government programs for health care, education, and the environment.
Option Elimination -
Here are the different usages of "while."
While + subject + verb: This is the most common usage of "while" to introduce a contrasting clause. For example: "While I was studying, my friend was watching TV."
While + gerund or noun phrase: "While" can be followed by a gerund (verb form ending in -ing) or a noun phrase. For example: "While running, he tripped and fell." or "While on vacation, they visited several museums."
While + prepositional phrase: "While" can be followed by a prepositional phrase. For example: "While on the bus, she read a book."
While + adverb or adverbial phrase: "While" can also be followed by an adverb or adverbial phrase. For example: "While quietly, she entered the room."
I shared these usages (other than to familiarize ourselves) to share a perspective that you please don't get stuck on the "while" issue. In this case, we can conveniently fill in the verb from the earlier clause (same subject and a verb of being), which means "they denounce big government while at the same time (they are) supporting many specific government programs for health care, education, and the environment. But the other options have much more serious issues. "While" is used as a conjunction to convey contrast; as long as it does that, we are good. There is no rule that "while" has to have a clause only after it.
(A) they denounce the big government, saying government is doing too much and has become too powerful, while at the same time supporting - ok
(B) they denounce big government- they say that government is doing too much and has become too powerful-but supporting at the same time - Before "but," we have a clause, and after "but," we have a phrase. Wrong.
(C) they denounce big government, say that government is doing too much and it has become too powerful, while they support at the same time - They denounce (verb), say (verb) - The intent here is not that they denounce and they say - the intent here is to describe adverbially how they denounce the government. Moreover, even if you look from another perspective, we can't miss out on "they" in one and use it in the other two.
(D) while they denounce big government, saying that government is doing too much and has become too powerful, at the same time supporting - There is no independent clause here after the semicolon (;) which is used to join two independent clauses.
(E) while they are denouncing big government- they say that government is doing too much and it has become too powerful-supporting at the same time - There is no independent clause here after the semicolon (;), which is used to join two independent clauses. The part between the dashes is just additional information.