The inferences that can be made about the grand-vizir’s motivations for speaking with the king in private are:
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He wants the king’s complete attention. - Speaking in private allows the grand-vizir to convey his message without distractions.
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He does not want anyone to warn Douban. - By speaking privately, the grand-vizir ensures that no one can alert Douban to the scheme being plotted against him.
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He is afraid of someone pointing out the error of his false accusations. - The private conversation minimizes the risk of others questioning or disputing his claims against Douban.
The other two options, "He believes that the king will shame him in public" and "He is embarrassed to publicly question Douban’s loyalty," are less applicable based on the context, as the grand-vizir seems more focused on manipulation and deceit rather than on concerns of shame or embarrassment.