Beckett's Scrabble-winning word "Quixotic" is an adjective referring to one who acts in a deluded manner similar to Don Quixote, the lead character in the Miguel Cervantes novel of the same name.
Beckett's score using "quixotic" with two triple word scores, one double word score, and the other 1-point word score plus 50 points for using all her letters. Castle only gives Beckett 314 points. The correct total should have been 356.
In the scene where Esposito invites Castle along for bacon brownies, Detective Sullivan's name plate is sat on the desk behind Beckett, indicating he still works there (albeit unseen since 6x04, "Number One Fan").