hp laptop broken screen, how to enter bios in external monitor

I pressed F10 at boot

Do you mean just F10, or Fn/F10? If you hold down the Fn key, each time you press F10 the display should cycle through Internal alone, Both, External alone. The function may or may not work before the POST is completed.

Since you don't appear to be familiar with its use, I'm guessing you've simply chosen the external monitor to be the primary display within Windows when it was detected, which indeed doesn't alter which screen the POST and BIOS screen will appear on.

You could experiment with an HDMI connection, but I'm not sure if it would change how the laptop's own startup screen is displayed.

I am not sure an external monitor will work with the BIOS settings.

I am working with a buddy's Dell laptop that has the backlight on the screen died. I have the laptop set to mirror to the external display that is connected by HDMI (only video out option on the laptop). It retains that setting when rebooting and after the laptop has been powered down and then powered backup. It even retained that setting after I "reset" the laptop to factory conditions (it, however, DID NOT display on the external monitor during the Dell reset process...only on the laptop screen).

Just for the heck of it after seeing this post, I rebooting the laptop and booted into the BIOS. Nada on the external monitor. But, I could see the BIOS on the laptop screen with the lights off and the flash light held at an angle to the laptop screen.

I cannot say if all laptops will act this way, but I kind of suspect that they will. You would need some sort of driver to load likely to be able to make use of the external output and that likely does not happen when you just boot to BIOS.

Oh, and I should note that not all laptops will just "toggle" through the various display modes with the Function key approach. The Dell I am working on does not. It brings up a menu where you then use the arrow keys to pick which display mode you want. Took me a bit to realize that was the case since the screen's backlight was dead...since I could not see the menu (or anything else on the screen), I did not know it was there. And Dell, in their infinite wisdom, does not have an actual user manual/guide for this model laptop...just a service guide (which will come in handy if my buddy decides it is worth replacing the LCD panel). So, I could find no description telling me about the menu. I just decided to try the display mode button while the lights were out and using the flash light.

Yes, I already acknowledged that if the system won't continue using the external monitor across a warm boot, and there is not a hotkey monitor selection that is preserved across a warm boot, it may not be possible.

Maybe we are not having a meeting of the minds in terms of "across a warm boot". As I said, the Dell will maintain the use of the external monitor when it is rebooted or even if it is shutdown. In other words, I do not need to constantly reset the display mode to use the external monitor. It just does not work doing anything outside of Windows...only the internal screen does. Not sure if that is what you mean by "across a warm boot" or not.

Dell being Dell, they may also have an unfortunate selection mechanism, and I don't have access to a Lenovo to check, but all the laptops I've had, Compaq, HP, ASUS and my current Toshiba cycled just on the Fn key combination. TheGreatOBO will need to let us know what results occur.

My experience in the past with other Windows laptops has been the cycle effect as well (this has included HPs, Compaqs and Toshibas). This also includes Dell laptops in the past if my memory serves me well (and it might not...it has been a LONG time since I put a Dell laptop in the external display mode).