I bought some new polished aluminum wheels 4 months ago and unfortunately neglected them a little bit and I went to clean and polish them today and there is a bunch of white spots I can't get off. I am wondering if there is anyway to get back the original shine and get rid of these spots?

 

those white spots are probably oxidation and they won't just wipe off. you will need to get a rotery tool with a buffing wheel and some high grade aluminum polish to get them shiney again.