If the laptop was shipped with Windows 8.1, then it most likely has Windows 8.1 product key embedded in motherboard memory. Laptops that have embedded product keys come with small Windows sticker at the bottom with Windows logo only. If you have a large sticker with product code printed on it, then the key isn't embedded.
The embedded key can be extracted using RWEverything. Click the ACPI Table button, then MSDM tab. The product key will be shown at the bottom.
If you don't have embedded product key and the key on the sticker is unreadable, then at this point you won't be able to retrieve original key - when Windows is upgraded, it generates a new product key that replaces the original one and can't be used for fresh installs.
Your options are:
Reset Windows 10. This version of Windows has built-in reinstall feature. Click:
Start → Settings → Update & Security → Recovery → (under Reset this PC) Get started
Windows will ask you which files you'd like to keep, then it will take some time to reinstall. Finally you'll be left with a fresh copy of Windows 10 without any licensing issues.
Try to clean install Windows 10. Download Windows 10. The official tool will write it to a thumb drive of your choice. Boot using that drive and try to install Windows.
If you have embedded product key, it shouldn't ask you to enter product key at all - it's likely that installer will recognize and accept Windows 8.1 product key. If it doesn't, see point 3.
If you don't have embedded product key, but a large product key sticker, you'll have to enter the product key manually. It will most likely work. If it won't, see point 3.
Install Windows 8.1 and upgrade to Windows 10. Windows 8.1 can also be downloaded from Microsoft. It will read the embedded product key if available or will ask you to enter it. After clean installation prepare a Windows 10 installation thumb drive and run it in Windows 8.1. It will start the upgrade.
If you have a Windows ISO and you want to write it to a thumb drive without official tools (for whatever reason), use Rufus. Microsoft tools prepare universal thumb drives, but Rufus forces you to choose between UEFI and BIOS styles (for complicated, subjective reasons). You'll have to figure out if you're using UEFI boot or BIOS/CSM boot. Windows supports UEFI boot only on GPT hard drives and BIOS/CSM boot only on MBR hard drives. Your current partitioning scheme (MBR or GPT) can be determined using built-in Disk Management tool:
right-click disk's label → Properties → Volumes tab → Partition style