Nvidia drivers dual screen Debian 12

Hi !
I’m currently on Debian 12, and getting a dual screen to work is very difficult for me (though I’m a computer science student…)
My setup is as follows:
Processor: Intel core i5-8300h with intel uhd 630 graphics
GPU: Nvidia GTX 1050Ti
I’m on a laptop PC, and the HDMI cable for it (which allows me to have a display on an external screen) is that of my NVIDIA graphics card.

Using nvidia-detect, this is what it shows me:
root@superlinux:~# nvidia-detect
Detected NVIDIA GPUs:
01:00.0 VGA compatible controller [0300]: NVIDIA Corporation GP107M [GeForce GTX 1050 Ti Mobile] [10de:1c8c] (rev a1)

Checking card: NVIDIA Corporation GP107M [GeForce GTX 1050 Ti Mobile] (rev a1)
Your card is supported by all driver versions.
Your card is also supported by the Tesla drivers series.
Your card is also supported by the Tesla 470 drivers series.
It is recommended to install the
nvidia-driver
package.
root@superlinux:~# xrandr
Screen 0: minimum 8 x 8, current 3440 x 1440, maximum 32767 x 32767
HDMI-0 connected primary 3440x1440+0+0 (normal left inverted right x axis y axis) 800mm x 335mm
3440x1440 59.97 + 74.98* 29.99
2560x1440 59.95
1920x1080 60.00 59.94 50.00 60.00 50.04
1720x1440 59.98
1680x1050 59.95
1440x900 59.89
1440x576 50.00
1440x480 59.94
1280x1024 75.02 60.02
1280x960 60.00
1280x800 59.81
1280x720 60.00 59.94 50.00
1152x864 75.00
1024x768 75.03 70.07 60.00
800x600 75.00 72.19 60.32 56.25
720x576 50.00
720x480 59.94
640x480 75.00 72.81 59.94 59.93
root@superlinux:~#

Here is the problem:
either my GPU allows me to have full performance on my external screen, but the internal screen of my laptop PC is not detected graphically (black screen with white dash at the top left) => in case I have /etc/X11/xorg.conf file in my machine.
either, if I delete the xorg.conf file, my two screens are functional but my external screen does not exceed 30 Hz (even if I put the maximum frequency in the parameters)

I use KDE Plasma 5.27.5 and a 6.1.0-9-amd64 kernel (all without secure boot)
I don’t understand how to solve the problem, I would just like the compromise between the two: my full performance external screen as well as my internal screen detected

Thank you in advance, best regards,
Nael

So…
I finally found it: I simply did an nvidia-xconfig --prime, and that fixed all my worries!
Thank you anyway !