how do you save a scanned document on a mac?

If you are using Image Capture to scan the image, there should be a dialog window where you can choose the name and where to save it to - I generally choose the desktop so I can see the icon immediately. Can always file it into a folder later.

Well, I don't have an HP, but Image Capture is in your Applications folder - I have an Epson and using the native Image Capture works a lot better than trying to keep up with the Epson scanner drivers. Does your HP not have an app you scan with (which would bring up a save dialog window)? If not, open System Preferences > Print Pane and see if your HP is listed on the left. If not, hit + to add a printer. If it's listed, hit the Scan button and that should automatically launch Image Capture.

yet another example of unintuitive nonsense from Apple

the scanner rescanned twice and tehn didn't save it anywhere

how can developers release stuff in this state?