Recent graduates often demonstrate poor writing skills, hiring managers say.

Subjects is graduates
Verb is often
Object is skills

Please check my answers for subject, verb and objects.

The subject and object are correct; the verb is wrong.

Hint: often is NOT something you can do.

Verb is demonstrate

yes

.... but I agree that it is usually grammar :)

Your answers for subject, verb, and object are correct:

Subject: graduates
Verb: often
Object: writing skills