From Longman Dictionary of Contemporary Englishcould/can you possiblycould/can you possiblyspokenASK FOR something/ASK somebody TO DO something used when making a politerequestCould you possibly close that window? →possibly
Examples from the Corpus
could/can you possibly• How can they possibly control such success?• But what on earthcan he possibly have been trying to prove?• What conversationcould we possibly have had there, then?• But what else could she possiblyhope to see? she wondered miserably.• How could she possibly know, since he had not set eyes on the girl?• What can I possibly say now?• What could she possibly tell him, that he would want to hear?