Formatting won't do what you want, really, but re-Partitioning it will. I think there's third option that is best.
I would recommend that you Zero-Fill that old HDD. Why? Well, that operation writes just zeroes to EVERY sector of the HDD, thus over-writing all data so it is very hard (impossible for an amateur) to recover any old data. That in itself may not matter much to you. But here is what is useful. When you write to a HDD sector, an internal operation that is part of the HDD's own programming will then read back what was written and evaluate the signals. If they are all strong and give correct data, it simply goes on the the next task. BUT if the data are wrong, or the signals from the heads are weak, the system attempts to re-read that data to get a good copy (if possible), then marks that Sector as poor and NEVER to be used again, and replaces it with a good sector from a stock of spares. You see, at the time that the HDD is first Low-Level Formatted at the factory (you cannot do this at home) the HDD has more Sectors than needed for its specified capacity. The known-good un-needed Sectors are logged in a record on the HDD for its own internal use. Having replaced the bad sector with a good spare, the system then will write the recovered data (if it could do that) to this sector. THEN it proceeds to the next operation. So, if you do a Zero Fill, EVERY Sector gets this treatment, so EVERY Sector is tested and ALL weak or failed ones get replaced. When the process is finished, there are NO "Bad Sectors" visible to any outside system, so your OS never knows anything about all this. The HDD appears to be in perfect condition and totally empty, just like brand new from the factory.
Well, not quite. "Brand new" means it has all those spare known-good sectors. But part of the process of testing and self-"fixing" (not really, because it cannot repair a true bad sector, only replace it) is that it also keeps a record of the number of replaced Sectors, and a "brand new" unit has none. There is a limit to how many of these replacements can be done, of course, and that is part of the data tracked by the SMART system. If you ever get a warning message from the HDD SMART system that the number of failed and replaced Sectors is too high, it means two things: your HDD is still working OK, but sometime in the near future it may not have any more good spares to use and cannot "fix" itself; and, the rate of failure may increase, so you should get busy and arrange to replace this HDD before it does actually fail.
So, if you do a Zero Fill it will erase all old data and do a complete test and replacement of weak Sectors. Then you can examine the SMART data from the HDD and verify it still is healthy. THEN you can re-Initialize the unit, knowing it has no faulty Sectors available for your OS to use, and it is still reliable.
One free utility that can do this is DBAN. You also can get free HDD diagnostic suites from the maker of your old HDD. Since it is a Seagate product, download from them their Seatools utility.
Doing a Zero Fill is a LITTLE bit of risk. It destroys every bit of old data, so you MUST be sure what HDD you run it on!! For that reason, personally I like to disconnect ALL other drive units and leave ONLY the one I'm working on attached so I cannot make a mistake. Of course, that means I don't have an Operating System to use. For that reason, I prefer to download the Seatools for DOS version. It is an .iso file, and that means it is a complete image of a bootable CD that you can burn to your own CD-R disk. In fact, you must - you cannot run this file from a hard drive. You download the .iso file. Then you need to get a utility able to burn that file to a CD-R disk - Nero is one such. Burn the CD-R. It is now a bootable disk you work from. Place the CD in your optical drive. Shut down, open and disconnect other drives leaving only the old HDD you want to clean connected. When done, boot into BIOS Setup immediately and go to where you set the Boot Priority Sequence. Set it to boot from the optical drive and nothing else. SAVE and EXIT, and the system will boot from the CD. It loads a mini-OS into RAM and creates a RAM disk for you to store stuff in if you want, then starts up the utility package and presents you with a menu of operations. Verify that it is set to work on the HDD you WANT it to, and not another. Set it to do the Zero Fill and be patient - it takes several hours on a large HDD. When it is done you get back to main menu, where you can see the SMART data displayed. When you are finished, remove the CD and shut down the machine. Disconnect power, open it up, and re-connect all the drives you have to the required ports. Close up, connect power, and boot again into BIOS Setup where you re-specify your desired Boot Priority, and SAVE and EXIT. When the machine has finished booting into your Windows OS, use Disk Management to Create a New Simple Volume on the cleaned-up old Seagate unit. Most default settings are OK. Make sure it uses all of the space available, and you do NOT need it to be a bootable drive unit. I'm guessing you are using the older MBR Partitioning system, but you CAN chose GPT if you wish and if you have the right version of Windows.
Some people are not comfortable doing this that way, and prefer to use the Seatools For Windows version. This can be downloaded and installed on your SSD as a standard Windows application. Then you run it under Windows, but this is REALLY where you MUST be sure when you launch it that it is set to operate on the old Seagate unit you want to wipe, and NOT another. (It will only operate on ONE unit.) Again, after it has finished you back out and use Disk Management to Create a New Simple Volume on the cleaned-up Seagate HDD.