He was busy doing his homework

“He does [his] homework” is correct, for simple present tense: meaning, he does homework habitually. More common, since usually people describe what they are currently doing for such things, is: “He’s doing his homework”. That means that he is currently, at this time, doing his homework.

It's a common and handy word that is used frequently, even in places where its un-contracted meaning "was not" would sound strange. Nobody ever says "Was not he doing his homework?" but because of custom and habit, English speakers freely say "Wasn't he doing his homework?". It's a quirk of the language.

Perhaps the prospective Conservative candidate would be best employed doing his homework, rather than making erroneous statements in the press. He spends about 10 hours per week doing his homework, communicating with his professors and fellow students via the Internet.

Doing your homework before you make your request will help you to have a confident manner. examine thoroughly the details and background of a subject or topic, especially before giving your own views on it.