[Gateway] Can I recover my Original OS [Windows 8] with just my machines Serial Number??

[Mosdyne]

My Grandfather gave me a computer he didn't want, and it came with windows 8. At the time I wasn't a big fan of it, so i removed it and installed Windows 7. I went the custom route on the installation and erased EVERY partition. So I'm sure the place that stored the Product key for windows 8 went with it. The sticker was removed sometime before it was given to me as well, but the machine has a list of specs on the side of it:
Windows 8
AMD Dual core
500gb HDD
6gb of DDR3 Memory
HDMI compatible

So i was hoping if i got in contact with someone at Gateway, they'd be able to tell me my product Key attached to the machine via Serial Number. I tried live chat on their support site but the support warranty expired, so i couldn't even speak with anyone from the site. Did I screw myself? because i really want to get a copy of Windows 10 without paying 100 something dollars for it.

 

[popatim]

Your product key is built into the laptops firmware. Just contact Gateway and order your factory restore disk set and when it installs the OEM version of win8 it will read the key directly from the laptop.