The main issue with the underlined portion of the sentence is that "complexing" is not actually a word. The correct answer will appropriately change the word into the best form of "complex" to fit the sentence. To describe the elements best involves making the word into the phrase "more complex."

This sentence demonstrates a comma splice, in which a comma connects two complete sentences. Comma splices are incorrect, as the independent clauses should either remain separate and be treated as complete sentences or be linked by a semicolon or a conjunction. "Coffee can be expensive, but also needing it to stay awake" lacks a subject in the second clause. "Coffee can be expensive; to stay awake" illustrates an incorrect use of a semicolon, as the second clause is not a complete sentence. "Coffee can be expensive, however, the reason I need it is to stay awake" repeats the comma splice of the original sentence. "Being expensive, I need coffee to stay awake" is an example of a dangling modifier, where "being expensive" incorrectly modifies "I" instead of "coffee." Therefore, the correct answer must be "Coffee can be expensive; however, I need it to stay awake," which displays the correct use of a semicolon to connect two complete sentences.

A complete sentence must include a subject and a main verb. Because the original sentence contains two modifying clauses but no main verb, it is actually a sentence fragment. The incorrect answers all contain modifying clauses but lack a main subject-verb relationship. The correct answer therefore must be, "Historians of the colonial Caribbean often focus on the issue of slavery, as its legacy continues to be a major theme in the region." This sentence contains the subject "Historians" and the main verb "focus," and it is therefore a grammatically complete sentence.

In this sentence, the verbs "to fish," "to hike," and "to go" must use parallel structures (the verbs must be in the same tense). The option "fishing, hiking, and going to the beach" is the only one in which all three verbs are in parallel structures, making it the correct choice.

Because the underlined portion of the sentence is an introductory phrase it must be followed by a comma. The incorrect answer choices omit the comma, change the meaning of the sentence, don't agree with the sentence, or are excessively wordy. Thus, because the given sentence separates the introductory phrase with a comma and is grammatically correct, no change is necessary.

The original sentence is an example of a dangling modifier. The preceding clause, "Happy that the long winter was finally over," refers to John, so John must be the subject of the second clause. John's coats are incapable of being happy that winter is over, so "John's coats" cannot be modified by that first clause.

"An account of the professor's research and teachings, which are described at length in the autobiography of his former student." - As it is presented in the question stem, the sentence is incomplete. It contains only a subject and is missing a predicate; "which are described at length in the autobiography of his former student" is a participial phrase describing "an account of the professor's research and teachings." The "account" doesn't do anything in this sentence or have anything done to it.

"An account of the professor's research and teachings being lengthy described in the autobiography of his former student." - This sentence contains two errors. One error is its use of the adjective "lengthy." "Lengthy" is used like an adverb to modify "described," but adjectives can't modify verbs, so this sentence is incorrect. The sentence's other error is in its use of the word "being," which does not make sense. To be read as a present progressive verb, "being" would need to be preceded by "is." Even if "being" were read as beginning a participial phrase, the sentence would remain incomplete.

"The professor's student, in his autobiography, lengthy describes an account of the professor's teachings and research. " - This sentence reproduces the "lengthy" error discussed in the previous answer choice, and it also contains an ambiguous pronoun. "His" could refer to either the professor or the professor's student.

"The professor's student, whose autobiography describes at length an account of the professor's research and teachings." - This sentence is also an incomplete sentence; it consists of a subject and lacks a predicate. The participial phrase "whose autobiography describes at length an account of the Professor's research and teachings" describes "The professor's student," but the sentence lacks a verb to tell us what "the professor's student" does in the sentence.

"An account of the professor's research and teachings is described in great detail in the autobiography of his former student." - This sentence has no errors.

"Young drivers in my grandfather's town often make the mistake of driving through red lights, they also are known for ignoring speed limits. " - This sentence fails to properly separate two indepdendent clauses, creating a run-on sentence error.

Young drivers in my grandfather's town often make the mistake of driving through red lights; also known for ignoring speed limits." - This sentence includes improper semicolon usage.

"Young drivers in my grandfather's town often make the mistake of driving through red lights, additionally they are also known for ignoring speed limits." - This sentence also contains a run-on sentence error. "Additionally" and "also" are also redundant when used in the same clause.

"Young drivers in my grandfather's town often make the mistake of driving through red lights, the other is for ignoring speed limits." - This sentence has two errors. It is a run-on sentence, and it is ambiguous what the "other" represents.

"Young drivers in my grandfather's town often make the mistake of driving through red lights; additionally, they are known for ignoring speed limits." - This sentence has no errors. 

"Having difficulty swallowing, Vitamin C are a supplement children don't usually take." - This sentence contains two errors. There is a subject-verb agreement problem between "Vitamin C" and "are"; since the subject is singular, the verb shouldn't be plural. As the sentence is framed now, "having difficulty swallowing," while true of "children," could easily apply to the "Vitamin C" itself.

"Vitamin C are a supplement children don't usually take, because they have difficulty swallowing it. " - This sentence contains two errors. There is a subject-verb agreement between "Vitamin C" and "are"; since the subject is singular, the verb shouldn't be plural. Additionally, there is an uneccessary comma after "take."

"Vitamin C, a supplement children don't usually take, having difficulty swallowing it. " - This sentence contains a verb tense error. "Having" both indicates the possessive and is in the wrong tense.

"Children don't usually take the supplement Vitamin C, having difficulty swallowing it." - This sentence contains a verb tense error. "Having" is in the wrong tense.

"Children don't usually take the supplement Vitamin C because they have difficulty swallowing it." - This sentence has no errors.

"It ought to be your mother with whom you cut the cake, not I." - This sentence contains one error. At the end of the sentence, "I" constitutes improper pronoun usage. It should read "me."

"It ought to be she, your mother with whom you cut the cake, not me." - This sentence contains a comma error. The interrupting phrase "your mother" was not correctly enclosed with commas.

"It ought to be your mother, she with whom you cut the cake, not I. " - This sentence is unnecessarily and confusingly worded. "she with whom you cut the cake" is a very awkward and confusing way of framing the sentence. Additionally, at the end of the sentence, "I" constitutes improper pronoun usage. It should read "me."

"It ought to be your mother with who you cute the cake, not me." - This sentence contains one error. "Who" is an incorrect object pronoun to use with "mother." In this situation, "whom" is correct.

"It ought to be your mother with whom you cut the cake, not me. " - This sentence has no errors.