I'm using an HP Pavilion with A8 APU and Windows 10 Home (upgraded from 8.1), bought about 18 months ago. The entire time I've owned it I have tried -- and failed -- to install any of the most popular distros in dual-boot. At the moment I'm running the Live CD (actually flashdrive) of Linux Mint 17.2 in order to write this post.

Installation always seems to go well -- /, swap and /home partitions are created and files are installed -- but when I reboot the computer fails to find GRUB and instead boots Windows normally, as if GRUB isn't there at all. UEFI is in legacy mode, secure boot is disabled; these are the same settings I used to install Windows 10.

Does this particular laptop require installing GRUB to a non-default location? Is there something else I need to do in order for UEFI to find and use GRUB?

I'm not a Linux newbie, and fortunately I have an older desktop that is Linux-only and works just fine. But my two desktop boxes are in my office and the laptop is usually in my living room. I don't want to get rid of Windows 10 on the laptop, since it also serves as a "guest" machine for computer-challenged visitors -- and in all honesty, I happen to like Windows for some things. Any help/advice you can offer will be appreciated.

It is eithe UEFI or Legacy, I believe the Legacy option is usually called CSM. I don't use UEFI myself but from what I have read, you need both systems installed using UEFI or BOTH systems installed using Legacy. Do you have an EFI partition on the disk? Do you have windows as well as Linux files in that partitions directory? Google "boot repair ubuntu" then go to the site using the Mint installation medium and download and run it. Do NOT try any repairs but select the option to "Create BootInfo Summary" and post the output here. Some members familiar with UEFI should be able to help when you have posted more detailed info. If not, you could try the ubuntuforums as there are a lot of posts there on this problem with solutions.

Correction to my OP: the laptop uses what looks like traditional BIOS, although oddly enough it refers to UEFI -- too many computers, I confused myself.

I've tried installing GRUB to Windows boot partition, Linux / partition and EFI partition, all with no success; on reboot, the computer still fails to find GRUB and boots Windows 10 normally. BIOS is set to Legacy mode, Secure Boot off -- these are the same settings I used to install Win10. I have tried resetting to EFI mode with Secure Boot on, but it seems to make no difference. I'm considering keeping this laptop Windows-only and buying a used older laptop on eBay for Linux-only work. Or maybe moving the TV to my office (where I have a desktop that runs Linux beautifully) so I can web-surf and channel-surf at the same time.

I'm starting to think there is something peculiar to the laptop's motherboard and/or BIOS design that blocks dual-boot, perhaps a deliberate limitation on the part of HP to simplify tech support.

That doesn't make much sense.

From your live Mint system, post the output of:
If you have an EFI system partition then Windows is installed in EFI-mode and your GNU/Linux system will also have to be installed in EFI-mode for you to be able to use GRUB to select between the operating systems (otherwise you can only switch by using your firmware boot menu and toggling "Legacy" mode).

To do this, *disable* "CSM"/"Legacy" mode before booting the installer.

It is actually possible to install Ubuntu, Fedora or OpenSUSE with Secure Boot enabled if you wish.

EDIT: I have encountered UEFI firmware implementations that will boot a UEFI system in preference to a "Legacy" (non-UEFI) system even if "CSM"/"Legacy" mode is enabled.

We can't help if we don't have hard data - bootinfo plus thatht efi as well.
Maybe that should be usually.
I have an older UEFI laptop where the Win10 preview wouldn't install to a gpt disk, and always installed in "legacy" mode. Even over-wrote the first sector (aka MBR) on a gpt disk without erasing the gppt header. Caused no end of grief with subsequent Linux installs.

The UEFI standard unfortunately isn't.

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