"my master has no concern about his household with me in charge." This can be written in positive form. Alternate translation: "my master trusts me with his household" (See: [[rc://en/ta/man/translate/figs-doublenegatives]])
When something is "put under someone's care," it means that the person is responsible for its care and safe-keeping. Alternate translation: "he has put me in charge of everything that belongs to him" (See: [[rc://en/ta/man/translate/figs-metaphor]])