As I run into issues on systems, and solve those, I thought it would be helpful to post them here.
Many folks who use different HP All-in-One (AiO) printers are finding they can no longer open the HP Solution Center to do scanning or other tasks. And the HP Image Monitor taskbar icon also will not open the Solutions Center.
The problem is that Adobe Flash Player 10 has been updated so it is no longer backwards compatible with older Windows versions, and apparently no longer support some softwares that are. If you happen to be good about updating (which is commendable) then you will likely have Flash Player 10, and no HP Support Center opening.
The solution in this case is to go
here and download Flash Player 9. Then go to Add/Remove Programs and click and uninstall Adobe Flash Player 10 ActiveX.
Click the downloaded install_flash_player_9_ax.exe file and allow that to install Flash Player 9.
Reboot, and check the HP Solution Center then. It should now open without problems.