Stupid SharePoint 2010 Error:The Installation of this Package Failed

If you are trying to install SharePoint 2010 and you get the error message shown in the following image, you will need to redownload the installation file and try again.

I wasted an entire afternoon trying to figure out why I was getting the error above. I had installed all of the prereq's and my Server 2008 R2 was up-to-date. The executable would not even fully extract, so there was no error log or anything to help me troubleshoot. I hope this post saves someone some time.