HP pavilion a345c has graphics card (Nvidia GeForce MX 440) integrated into the motherboard - a fairly common arrangement.

When I install a 2nd (Nvidia) graphics card in the proper slot the system recognizes that card instead and does not acknowledge any graphics adapter capabilities on the motherboard.

How to enable both and get dual monitors to work ?

When I install a 2nd (Nvidia) graphics card in the proper slot the system recognizes that card instead and does not acknowledge any graphics adapter capabilities on the motherboard.

Also fairly common. You need to look in the BIOS Setup Menu. There where be a setting that enables the on-board, or shuts off the auto-change to a card. I don't know the wording, just function.

That said, if you could use a card that supports two monitors, that would be MUCH better. The card most likely would have a better GPU than the on-board, but perhaps more importantly here is that the card will have its own dedicated RAM that has been tweaked for graphics processing.

With the on-board, a substantial chunk of system RAM is stolen... errrr, I mean "shared" for use by the on-board graphics. So when you switch from on-board to a card, not only do you get a better graphics solution, you get a little RAM boost in the process because that previously snagged chunk of RAM is released back to the OS and CPU for their use.

So, as a dual monitor user for many, many years, a separate card that supports your monitors would be a better option than using on-board and a card.

Also, while Windows supports multiple monitors, many programs have problems. And even the latest operating systems sometimes have little issues, and seem to get confused occasionally as to which monitor is primary and which is secondary. So a multi-monitor utility is almost a necessity. My favorite is UltraMon – not free, but worth it. Others have suggested the free DisplayFusion, but I have no experience with it.

the BIOS seems to be either-or for the display adapter

Then I suggest a card with dual-monitor support.