MAGOOSH OE:
A question about Samuel Tilden and the controversial 1876 Presidential Election.
Choice (A) is a grammatical correct structure known as an absolute phrase. This choice is idiomatically correct and logically clear. This is a promising choice.
Choice (B) commits an error involving the structure "with" + [noun] + [participle]. As discussed in this blog, this structure is fine for descriptive detail, but for action by a different actor, as we have in this sentence, this structure puts too much action inside a prepositional phrase. For these reasons, (B) is incorrect.
Choice (C) makes a logical mistake. In the situation described, Tilden's supporters acquiesced in exchange for somebody else giving the order of withdrawing the Federal troops from the South. In other words, Tilden's supporters had no direct dealings with the Federal troops and certainly didn't exchange these troops for anything. Choice (C) is incorrect.
Choice (D) also changes the meaning. In this choice, the final part is stated as two separate fact: (1) Tilden's supporters acquiesced as an exchange (an exchange for what??) and separately (2) Federal troops withdrew. In this choice, it's not clear whether these two facts have any logical connection at all. This is very different from the prompt meaning. Choice (D) is incorrect.
Choice (E) also changes the meaning by change nature of the exchange; this phrasing suggests that all the action after the last comma were in exchange for something earlier in the sentence, and this is very different from the prompt meaning. Choice (E) is incorrect.
Choice (A) is the only possible answer.