After you've uninstalled Office 2010, do an Advanced Search for any remnants of program, that might still be lurking on your system. But before doing the Advanced Search, go into your C drive to manually search for the program, and if you find any program folders/files associated with Office 2010, manually delete them. (If the system doesn't let you delete the whole folder at one time, it may very well allow you to delete each file at a time. Or, you can sometimes fool the system by renaming the files and then deleting them.)
Then, when you've done all you can do manually, do an Advanced Search for those remnants of remaining program files, and delete them that way. (For some reason, the system seems to be more forgiving about deleting files this way, then uninstalling them.)
To get to Advanced Search, press the Windows key on your keyboard + F (or go into the Start Menu, and from inside the Search Pane, press F3).
That brings up the Search page - click the Advanced Search button to the far top-right, to open the Advanced Search Set Up page.
On the Set-Up page, check the 'Include non-indexed, hidden, and system files' option.
Under Location, select Computer, or Everywhere or C Drive, and in the Name box, fill in: Office 2010 (using the actual exact name of the software), or some text from within the program's folder. Click on the Search button, and if any associated files come up, just delete them.
After this first search is completed, to make sure there aren't still any remaining files lurking, do additional searches, using any other key words you think might be associated with the program – like, for example, the name Microsoft.
PS: A word of caution: Some files will undoubtedly come up in the search that are NOT associated with your Office 2010. If you're not sure, don't delete those files, because you might be deleting something integral to another file or application. You have to be very careful! However, if you do delete something by accident, fortunately, you can always recover it from the Recycle Bin when you catch your mistake.