Take a look at the sentence again:
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 |
| begin | begin(s) | began | begun |
Began, the form you chose, is the simple past form; you cannot use the simple past with auxiliary verbs like would have. Before you continue, review the rules for irregular verbs.
Go back to the sentence to try again.