What word best completes the request to speak to Mr. Brown?

The choice "Could" is appropriate in this context because it conveys a polite request or inquiry about the possibility of speaking with Mr. Brown. Using "could" suggests that the speaker is considering the option of speaking to him in a respectful manner, indicating a sense of deference to the listener's ability to facilitate that conversation.

In requests like these, "could" is commonly used to imply both capability and politeness, making it a suitable option for asking for permission or the opportunity to communicate with someone. "Would," while also polite, typically suggests a willingness or intention rather than the possibility, which makes it slightly less fitting here. Similarly, "should" implies an obligation or recommendation, and "might" conveys a sense of uncertainty or less direct politeness, neither of which aligns as closely with the request to speak to someone in this context.