expression, godan verb (-bu)
curry favour with; win someone's good graces
A set phrase meaning to actively try to get into someone's good books, often by flattery or kind acts. The verb 取り結ぶ here adds a nuance of establishing or building a relationship.
彼は上司の機嫌を取り結ぶために、毎朝コーヒーを買ってくる。
He brings coffee every morning to curry favour with his boss.
あの政治家は有権者の機嫌を取り結ぶことばかり考えている。
That politician only thinks about winning the voters' favour.