Explanation
Correct answer: B.
Choice B is the best answer. The conventions being tested are the use of verbs to express tense and the use of verb forms within a sentence. In this choice, the past tense verb "became" is consistent with the other past tense verbs ("owned," "operated," and "was") used to indicate that, at a period of time in the past, Bass’s leadership resulted in her newspaper becoming one of the most influential Black-owned newspapers in the US. In addition, "became" supplies the main clause with a finite (tensed) verb, which is required to perform the action of the subject (in this case, "The Eagle").
Why the other choices are wrong
Choice A
Choice A is incorrect because the future tense verb "will become" isn’t consistent with the other past tense verbs used to discuss Bass and her newspaper.
Choice C
Choice C is incorrect because the present progressive verb "is becoming" isn’t consistent with the other past tense verbs used to discuss Bass and her newspaper.
Choice D
Choice D is incorrect because it results in an ungrammatical sentence. The nonfinite to-infinitive "to become" doesn’t supply the main clause with a finite verb.