The sentence is written as a compound sentence, one that joins two indpendent clauses together in one sentence to show their relationship. A compound sentence must be joined either by a comma followed by a conjunction, or a semicolon. The only answer choice that properly joins the two independent clauses is "the biggest difference in a project; the big things."

To clarify and simplify the sentence, you can use a semicolon to separate the independent clauses, and you can separate the items on the list with a comma.

Each of the clauses being combined in the sentence, "Jeff couldn't stand his precalculus teacher" and "she would always demean him in front of his peers" could function on its own as a complete sentence. This means that each clause is an independent clause, and because of this, the two should not be linked by just a comma; doing this creates an error called a "comma splice." 

Without adding any words, you could merge the two clauses using a period or a semicolon; alternatively, you could add in an appropriate conjunction immediately after the comma. "As" works as just such a conjunction here, since Jeff's dislike for his teacher is because of her demeaning treatment of him. "Though" will not work, because it would suggest that he disliked her in spite of her negative treatment of him. So, the correct answer is "as she would always demean him in front of his peers," making the complete corrected sentence, "Jeff couldn't stand his precalculus teacher, as she would always demean him in front of his peers."

A colon is the most appropriate punctuation to use when combining the two independent clauses present in the sentence because the second clause clarifies or elaborates the first clause. So, the correct answer is "As their spaceship landed, the aliens began emitting a screeching message regarding their intent: they desired to make peace with the humans."

The answer choice "Caligula was emperor of Rome during the first century; and he was one of the most brutal rulers in history" is incorrect because it contains an ungrammatical use of the semicolon. Two independent clauses can be combined using a comma followed by a conjunction, but not a semicolon followed by a conjunction.

The original text "Caligula was emperor of Rome during the first century, he was one of the most brutal rulers in history" and the answer choices "Caligula was emperor of Rome during the first century, he has been one of the most brutal rulers in history" and "Caligula was emperor of Rome during the first century, he has proven to be one of the most brutal rulers in history" are examples of run-on sentences and are therefore incorrect.

The answer choice "Caligula, one of the most brutal rulers in history, was emperor of Rome during the first century" is the only answer choice that avoids creating a run-on sentence and misusing a semicolon, and is therefore correct.

The two phrases in the sentence have slightly contradictory ideas, with the second one modifying the first. A conjunction is needed to show these differences in the two phrases. The correct answer is the one that uses "but" to separate the two parts.

The sentence is a complex sentence, featuring two complete thoughts put together to form one sentence. To appropriately join the two thoughts, they need to have a semicolon or a comma and a conjunction. "The cold weather can be quite difficult for young children, so companies offer special clothing for infants during the winter months," is the only correct answer.

The second part of the sentence is a dependent clause, one that cannot stand alone as its own complete sentence; therefore, it needs to be appropriately connected to the first portion of the sentence with the correct preposition. Of all the answer choices, only "hotly debated, after new information" does this.

This sentence contains a comma splice, an error in which two independent clauses (sentences that could each stand on their own) are improperly joined by a comma. When joining two independent clauses into a single sentence, one should use either a semicolon or a comma followed by a conjunction. Two of the possible answer choices employ a comma followed by a conjunction, but the answer choice that uses "but" is incorrect, because "but" should join independent clauses that contrast against one another, which the two independent clauses in question do not do. So, the correct answer is the answer choice that uses a comma and "so" to join the two independent clauses, because the relationship between the independent clauses is one of cause and effect, or reason and action taken due to the reason, not one in which the ideas contrast against each other.

The sentence is composed of two independent clauses—two complete sentences, joined to show their connection; however, the two are joined only by a comma, creating an error known as a "comma splice." Independent clauses should be joined by a semi-colon or by a comma followed by a conjunction (a word like "and" or "but"). To correct this sentence, you could either change out the comma for a semi-colon, or insert an appropriate conjunction immediately after the comma.

Let's consider the possible answer choices. The answer choice that makes no changes to the sentence can't be correct, since the sentence does contain an error. "Also" is an adverb, not a conjunction, so the answer choice that uses it as a conjunction cannot be correct. The remaining answer choices properly use "but," "so," and "although" as conjunctions, so finding the right answer becomes a matter of picking the conjunction that matches the logic of the sentence. "But" and "although" are used to indicate that the second sentence contrasts in some way with the first; this wouldn't make sense in our sentences, since they don't contrast with each other, but are related by causation: the first sentence explains the second. "So" is thus the proper conjunction to use, making the correct answer, "She was quite wary of making a scene, so she let the matter fade away with no comment."