Luna looks like a pain in the ass character, and she's supposed to be the protagonist. Selling the right of entry because she wanted to buy some bauble, prostituting her knight because she's strapped for cash and still wanting to be the king? Telling a potential vassal to lick her shoes? What other lows can she commit?
This chapter tried to pass her off as considerate, kind and talented, but yeah, not buying it. A considerate person doesn't whore out her servants or make her servant lick her shoes. She's just a dumbass gyaru who thinks the world revolves around her. Yeah, met those kind of girls before, all so selfish and a pain in the butt. In that sense, this character is realistic, because if you give gyaru bitches the right to become king, they'd likely act just like her.
I agree with Felicia, she should drop out. There's no way she can be a decent person. She's more a pain in the ass than this one, and this one at least has her heart in the right place.
And also, don't insult your teacher like that! Answer the freaking math problem.