Bob b and Oldartq,
Thanks to helpful hints by both of you and Bob Proffitt, plus serendipity, I succeeded in what I wanted to accomplish, namely, a DVI-D connection between my new Dell E2414Hr monitor and my HP Pavilion p7-1110 desktop PC, simultaneously with a VGA connection between my older (Samsung) monitor and my PC.
The help by all three of you was certainly valuable; the serendipity part is as follows:
On reading Bob b's first two statements above, I thought I should just check whether disconnecting the VGA cable between the Dell monitor and the PC, and keeping only the DVI-D cable between them proves the solution. (On two accounts, I was not very hopeful that that would solve my problem: (1) When trying to get the DVI-D connection between the Dell monitor and my PC working yesterday, I did not have the VGA cable connection too between them. The DVI-D connection was never successful (even when of course I used to choose the DVI-D option from the Dell menu). Then I had added the VGA cable connection because at least it always worked (with of course my choice of the VGA option from the Dell menu). (2) I had felt that having both DVI-D and VGA connections simultaneously should not at all matter since one of them would be "dormant" --harmless-- depending on my DVI-D/VGA choice from the Dell menu.
Very surprisingly, when, after reading Bob b's first two statements, I disconnected the VGA cable (and chose the DVI-D option from the Dell menu), the display instantly appeared on my screen!! I have absolutely no idea why the same connection did not at all work earlier.
Moral of the story: Life is full of mysteries; some of them are annoying to different extents (and sometimes downright most unfortunate); some of them produce happiness. When something annoying or "bad" thing happens, we have to try our best to find a solution using our wits, and further seeking, when necessary, help of some experts out there, who with a remarkably helpful mindset, provide help!
This CNET Forum is superb!
My thanks and regards to all three of you:, Bob b, Oldartq, and Bob Proffitt.
Chuck J