The sentence is written as a compound sentence, one that joins two complete thoughts together to show the relationship between them. A compound sentence always needs to be joined either by a comma and a conjunction or a semicolon. "[T]o the public, but they often" is the only answer choice that does this appropriately.

The sentence in question contains a comma splice, or two independent clauses connected by a comma. Independent clauses should be connected using a semicolon or a comma followed by a conjunction. We can ignore the answer choice that uses a parenthetical to enclose the second clause because it does not use a semicolon or a comma followed by a conjunction, and the resulting sentence is awkward. While "protestors, but they" and "protestor, so they" each employ a comma followed by a conjunction, the word "but" sets the two clauses in opposition to one another, which is not an accurate representation of the sentence's content, and the word "so" introduces a cause-and-effect relationship between the two clauses that does not work with the sentence's content either. So, the best answer choice is "protestors; they."

This sentence is a run-on sentence because it contains two independent clauses, "The quetzal is a tropical bird that is famous for its vivid plumage" and "it is also important in Mayan mythology." To correctly connect these two sentences, you can use a semicolon or a comma followed by a conjunction. (Writing the independent clauses as two separate sentences would also be a way to correct the error.) The only answer choice that correctly uses one of these methods is "The quetzal is a tropical bird that is famous for its vivid plumage; it is also important in Mayan mythology."

A sentence with two independent clauses must have both clauses be able to stand as complete sentences, and be connected to each other in meaning through a conjunction. "And many students studied with him," is the only choice that fits those parameters.

The answer choice "written by poet Langston Hughes, and even today, it is widely regarded as one of the most influential works" is correct because it results in a sentence in which two independent clauses are separated by a conjunction followed by a comma.

The original text and answer choice "written by poet Langston Hughes; and even today, it is widely regarded to be one of the most influential works" uses a semicolon incorrectly. Sentences can be combined with a comma followed by a conjunction or with a semicolon, but not with a semicolon followed by a conjunction.

The answer choices "written by poet Langston Hughes, even today, it is widely regarded as one of the most influential works" and "written by poet Langston Hughes, it is widely regarded, even today, as one of the most influential works" create run-on sentences and are therefore incorrect.

The underlined portion of the sentence stands oddly next to the preceding phrase, as it is in no way connected to the preceding phrase. The phrase needs to be changed so that it appropriately links with the first part of the sentence. "With vendors finding their profits increasing with each event" is the correct answer choice.

The underlined portion of the sentence is awkwardly and confusingly worded, especially as concerns verb transitivity. The way the sentence is written makes it seem as if the promoters want to attend more people instead of them wanting more people to attend any given music festival. To make the sentence clear and sensible, the words can be rearranged to make it more understandable. "Want more people to attend" is the best choice among the answers.

The correct answer choice is concise, fluid and unambiguous. 

In the question sentence, the word, showing, is out of order, and is not in the correct tense.

The option to rewrite this sentence should clarify, but not change, its meaning. The correct answer: "As we grow older, our thoughts and feelings on faith sometimes change during the holidays," allows the ideas to flow more smoothly without altering the content: each fact is still relayed (the time of year; things that are changing, etc.), but the tone is much clearer.