My maddening failure to move from Win 10 Home to Win 10 Pro

My pre-built Dell system came with Win 10 home. After my Windows exploded, I had to reinstall. Along the way I learned that I need Bitlocker. Since Home doesn't include it, I purchase an OEM install of Win 10 Pro.

And that has turned out to be the most horrific experience I've had in some time. I did a straight install and, at the end booted into Windows 10 home. Then I formatted the hard drive, reinstalled with my Win 10 Pro disc (always with that disc that I just purchased) and then... boot into Win 10 home.

I cannot, for the life of me, install Win 10 Pro on my PC. Right now I am creating a Linux bootable-USB because I need some way to see if there's a hidden... something on my hard drive. This entire experience has been exhausting and frustrating and I haven't a clue what else to do.

It turns out there were two recovery partitions that were withheld from me during the recurring installs. I blasted them out. Now the install is happening and I'm confident Win 10 Home won't re-emerge. Thanks.

Okay, so it turns out I'm back to where I started from. I have a fresh install of Windows 10 Home and when I am prompted to enter my product key, I am told it didn't work. However, I have all the relevant cardboard and purchased this from Fry's so I know it's not some piracy thing. What's my next step?

The problem is that you have windows 10 OEM. The OEM edition is intended to either be installed on a system that does not have a key embedded in the firmware, like your system has.. or to use the embedded key if present. since your key is present, and it’s a HOME key, it uses that.

if I were you, I would instead purchase a 10 pro upgrade, and sell your full install. You cannot upgrade from home to pro with an OEM full key

I tried that, yeah. But Oh wow. The devil's in the details. Well that was a waste of cash. Yeah, I'll have to find someone who wants this and buy a proper upgrade. Thanks Mystere.

Just to confirm it, @Mystere was exactly correct.