pay off the principal
Not to be confused with the shrewdly-prudent and virtuously-responsible practice of diligently settling your bills "up front and in full" and therefore carrying a zero balance (i.e., your "principle") on your credit card each month to avoid interest-fees, this term refers to the decidedly UNWISE and UNFAIR act of BRIBING (i.e., "paying off") the head-honcho at an establishment of learning (i.e., the "principal"), so that he'll kiss-buttingly bow to your outrageous stipulations and/or preferentially pull strings on your behalf, often to the detriment of others in the school.
Probably Ethan Couch's absurdly-indulgent father initially tried to pay off the principal so that his spoiled-a** Little Prince could be coddled and given unduly-preferential treatment, but then when this failed, he upped the ante by threatening to buy the entire school just so that Ethan could be tutored the way HE preferred him to be.