On July 1, 2012, Gamecock receives $5,800 from a customer for advertising services to be given evenly over the next 10 months. Gamecock credits Unearned Revenue. How do I calculate the unearned revenue for the adjusting entry?

Take your amount received of $5,800 times number of months of service provided divided by 10 months to get your adjusting entry amount. Your adjusting entry will be:
Debit Unearned Revenue for the amount calculated
Credit Earned Revenue for the amont calculated.

To get the remaining balance in Unearned revenue you simply take your beginning amount of $5,800 minus the amount debited your to Unearned Revenue account.