The HP Warranty Checker is really annoying, and it pops up randomly, and it kept popping up. After reading and testing some of the solutions like uninstall/reinstall to no avail, I decided to try a simpler less time-consuming cheat sheet method, and so far, so good. What I did was locate the path to the executable and renamed it. It's been over an hour, and it hasn't popped up yet.
C:\Program Files (x86)\HP\HP Support Framework\Resources\HPWarrantyCheck.exe. Renamed to C:\Program Files (x86)\HP\HP Support Framework\Resources\HPWarrantyCheckOld.exe The logic is it can't run what it can't see and because the system isn't dependent on it, it doesn't err because it can't find it. If ithe system does err because it can't find it in the future, I will simply rename it back to the original app name and look for another solution. But right now, this appears to work.
Update: Every time the system updates, it will add a new HPWarrantyCheck.exe file, so these steps will have to be followed for each system update.