I have a Seagate expansion drive, 1 terabyte, which I connect to a desktop computer. It has 400 GBs of data on it. When I use the "Safely remove hardware and eject media" icon, and put my cursor over the hard drive indicator, I get a circular "please wait" type indicator. This seems to go on indefinitely - I have waited 10 minutes. If I click while this is going I get the usual "safe to remove hard drive" message. Is it safe to do this or will I possibly scramble some of the data?
You need to find out what process is locking the drive. If it's not a process you've initiated, e.g. opening Explorer or looking at photos on it, try a tool such as the free LockHunter from Crystal Rich or an alternative.
Note that some of these applications may contain adware or malware, and check any new application with VirusTotal.
Once you find the locking process, you must decide if it's safe to ignore the lock, to kill the process or to perform full shutdown, after which the drive may be removed. Note that if the process is writing to the disk, simply unplugging the external HDD may damage the file system and cause data loss.
On top of what the guys pointed out about the possibility to corrupt the file system if something is writing data on your drive and you disconnect it: The "Safe Eject" function checks if all requests from and to the drive are completed and makes sure nothing gets lost in the cache of the drive. After confirming this, the OS writes a small amount of data on the drive indicating that nothing gets lost during the process of disconnecting the drive. The next time you plug the drive in your system or another system the OS is looking for that piece of data and if it does not find it it may ask you to reformat the whole drive in order for the OS to work with it and you may end up losing all your data from the drive.