Hello.
To avoid mixing up the computers, I will call the work copmuter a desktop and the home computer a laptop. Yes it is possable to use more than one screen at a time on a laptop. For example if you happen to have a smart T.V you can connect a laptop to the T.V with the Win+P button. Yes the view can be Duplicate[mirror] \_ Extend Will display a dragged "window" on extended device. If you need it to connect via wired cable.. > To use one monitor+laptop\_screen a laptop should have the display port icon label on the laptops USB port Or at minimum a HDMI port. Depending on the **monitor** connection if it uses a HDMI cable or a USB-C type looking cable Or even a dispay port cable that looks very close to a USB type-A cable but with a kinda L shape instead of instead of the \_\_\_ shape inside the port. Now if you needed 3 screens [ 1Laptop+2Monitor} then you will need a laptop that has two display icons labeled on the laptop. If you already own a laptop please provide some details about it. If you are shopping for a laptop you will want browse over at the gaming laptops with the fancy color changing keyboards.
I do not know what program you are using to view the desktop. I would think by defult it will show either both screens at once in a scaled down window in the viewer Or it will show the one screen at a time. Now if the program was a paid remote viewer like teamviewer paid addition I know you can view the desktop both monitors using that program.
Do you have a laptop already? If Yes, What is it?
Do you know what program you are tring to use for remote view?