drone on
To talk at length in a monotone. If you keep droning on like this, you're going to put your audience to sleep. A: "The professor's already been droning on about polynomials for an hour." B: "Oh, so that's why I lost interest an hour ago!"
drone out
To say something in a monotone. A noun or pronoun can be used between "drone" and "out." If you keep droning out information, you're going to put your audience to sleep.
drone on (about someone or something)
to lecture or narrate in a low-pitched, dull, and boring manner. The dull old professor droned on about Byron—or was it Keats? It was Shelley and, yes, he did drone on.
drone (something) out
to make a loud and low-pitched noise; to say something in a low-pitched and monotonous manner. The announcer droned the winning numbers out. She droned out the winning numbers.