Take another look at the sentence:
Jim would have __________ the difficult and dirty job of cleaning the slimy gutters if the blue sky, seventy-degree weather, and new titanium graphite driver had not summoned him to the golf course.
The sentence requires the past participle of begin, an irregular verb. Begun is the past participle.
|Verb||Simple Present||Simple Past||Past Participle|
Beginned, the form you chose, is a spelling error. Before you continue, review the rules for irregular verbs.