Explanation
Correct answer: B.
Choice B is the best answer. In this quotation, Dr. Proudie is described as "fondly flatter[ing] himself" that he has gifts that "must be sure to recommend him to a yet higher notice." In other words, he expects his skills to push him to greater fame and success. This implies an exaggerated sense of his own abilities, which matches the claim we’re trying to support.
Why the other choices are wrong
Choice A
Choice A is incorrect. This quotation doesn’t describe Proudie’s view of himself, nor does it paint him in an especially flattering light. Instead, by saying his mental powers and business skill are not "proved," it implies that he is actually dim-witted and bad at business.
Choice C
Choice C is incorrect. This choice describes Proudie’s closeness to power and importance, but it doesn’t show what Proudie thinks of himself. Proudie is not describing himself or his abilities here. The narrator is.
Choice D
Choice D is incorrect. While this quotation offers Proudie very mild praise, it doesn’t show what Proudie thinks of himself or his own abilities, which is what the claim focuses on.