Final answer:
Claudius's lines reveal his utilitarian and manipulative view of Rosencrantz and Guildenstern, only valuing them for how they can serve his own ends, as seen in the significant quotes from the play.
Step-by-step explanation:
The student's question relates to an understanding of King Claudius's attitude towards Rosencrantz and Guildenstern in Shakespeare's play. Considering the excerpt provided, several lines reveal Claudius's manipulative and utilitarian approach towards these characters. Particularly, the line "The need we have to use you did provoke / Our hasty sending." highlights Claudius's candid admission of using Rosencrantz and Guildenstern for his own ends. Another notable line is "I entreat you both / That, being of so young days brought up with him", which suggests Claudius's exploitation of their past friendship with Hamlet to serve his purposes. Finally, "That you vouchsafe your rest here in our court / Some little time" presents his seemingly gracious offer as a ploy to keep them close and under his control. These lines collectively encapsulate Claudius's scheming nature and how he views Rosencrantz and Guildenstern as mere instruments in his political machinations.