In the following sentence, the bracketed part is grammatically incorrect. Choose the option that best improves the sentence: The speed with which (these hate crime videos travelling) on social media frames a difficult challenge for law enforcement authorities.

Introduction / Context:
This question examines subject verb agreement and correct tense in a sentence about social media and law enforcement. The sentence describes how fast hate crime videos move across social media platforms and how that creates challenges. You are asked to choose the grammatically correct and logically appropriate form of the bracketed part so that the sentence reads naturally in standard English.

Given Data / Assumptions:

• Original sentence: The speed with which (these hate crime videos travelling) on social media frames a difficult challenge for law enforcement authorities.
• The subject within the bracket is plural: "videos".
• The sentence describes an ongoing, typical situation, not a single completed event in the past.
• The verb phrase must agree with "videos" and with the idea of something that happens repeatedly in the present.

Concept / Approach:
For habitual or general truths, English normally uses the simple present tense. We say "videos travel quickly on social media" to describe a general pattern. The structure "these hate crime videos travel" is therefore the natural choice. The word "these" agrees with the plural noun "videos". The form "travelling" in the original is participial and does not fit into the sentence structure as a finite verb. In the options, incorrect determiners ("this") or incorrect tenses ("travelled") disturb the meaning and the grammar.

Step-by-Step Solution:
Step 1: Identify what the clause needs: a finite verb that shows what the videos do.Step 2: Recognise that the situation described is general and ongoing, so the simple present tense is appropriate.Step 3: Match the determiner with the plural noun: "these videos" is correct, "this videos" is wrong.Step 4: Insert option A: "The speed with which these hate crime videos travel on social media frames a difficult challenge for law enforcement authorities."Step 5: Check for both grammatical correctness and clarity; option A satisfies both conditions.

Verification / Alternative check:
Read the sentence aloud using each option. With option A, the sentence flows smoothly and accurately reflects what we see in reality: hate crime videos move quickly online and that creates enforcement issues. Option B immediately sounds wrong because "this videos" breaks determiner noun agreement. Option C, using "travelled", sounds like it refers to a one time situation in the past rather than an ongoing challenge. "No improvement" is not acceptable because the original phrase "videos travelling" has no finite verb and leaves the clause incomplete.

Why Other Options Are Wrong:

• Option B: "this hate crime videos travel" uses the singular determiner "this" with the plural noun "videos", which is grammatically incorrect.
• Option C: "these hate crime videos travelled" shifts the idea into the past, which does not match the idea of a continuing challenge described in the rest of the sentence.
• Option D: "No improvement" leaves an unfinished verbal structure in place. The participle "travelling" cannot function as the main finite verb in this clause.

Common Pitfalls:
Students sometimes overlook agreement between determiners and nouns because their attention is fixed on the main verb. Another common confusion is between present simple and past simple. When a sentence speaks about a current pattern that creates an ongoing problem, the simple present tense is preferred. To avoid such pitfalls, always identify the time reference in the sentence (habitual, present, or past) and check that all the grammatical elements line up with that reference.

Final Answer:
The correct improvement of the bracketed part is these hate crime videos travel.