The 8862 is used to request the IRS reinstate one's qualification for a credit when the taxpayer has broken federal law by filing a fraudulent tax return and claimed a credit they did not qualify for. It does not pertain to just the AOTC either.
An undergraduate college student that was under the age of 24 on Dec 31 of the tax year generally doesn't qualify for the credit - especially in their first year of college since they still qualify to be claimed as a dependent on their parent's tax return. But the parent's of that student will probably qualify for it.
The HOPE credit died back in 2010 or there-a-bouts. It was replaced with the American Opportunity Tax Credit (AOTC).
