I was using two monitors. Each had both VGA and DVI connectors. Connecting the second monitor via VGA gave me occasional ghosting (i.e., a block of color would smear from left to right, slightly visible beyond where it was supposed to be). It appeared to be a VGA-related problem.
I guessed (maybe with the aid of some previous browsing) that the problem would go away if I used DVI or HDMI instead. I would have liked to connect both via DVI or HDMI, but the computer had only one VGA, one DVI, and one HDMI jack. I had a spare video card, but it too had just one DVI and one VGA jack. It looked like video cards with two DVI jacks or two HDMI jacks were expensive. (Later, prices on at least one such card had dropped to $70, though that was still more than I cared to pay.)
The motherboard did have an unused HDMI jack, so I bought a DVI-to-HDMI adapter. But the HDMI port was right next to the DVI port, on both the motherboard and the optional video card. The adapter was too big to fit there, if I was also using the DVI jack. The adapter had a female DVI connector, so I couldn't use it on the other end, at the monitor, where a male connector was needed.
It seemed that I needed an HDMI cable to run from the computer to the monitor, and a male DVI adapter to connect to the DVI port on the monitor. The question there was which kind of male DVI adapter I needed. Based on drawings at Wikipedia, there appeared to be at least a half-dozen possibilities. As elsewhere in life, it seemed the ideal solution would involve the minimum number of pins on the male side, and the maximum number of sockets on the female side. That would minimize the risk that there would be one or more male pins without a corresponding female socket, in which case no connection would occur.
All DVI configurations seemed to consist of a spadelike (flat, wide) connector at one side (let's say the left) and one or two roughly rectangular arrays containing 12 to 24 small round pins on the other side (i.e., in the center and toward the right). The DVI-D offered one ideal male configuration, insofar as its spadelike connector was not surrounded with four extraneous round pins. The Single Link option offered another ideal male configuration, with only 18 pins arranged in two clusters of nine in the center and on the right (rather than 24, as in the Dual Link alternative). Putting these two together, the most available male was the DVI-D Single Link: its spade side would match any DVI female, and the two sets of nine pins on its right side would match all DVI-I and DVI-D females.
The female DVI jacks on both the computer and the monitor offered a relatively agreeable DVI-D Dual Link connector, with the maximum number of sockets in the right-side cluster. They fell slightly short in not offering the extra four sockets on the spade side that a female DVI-I Dual Link connector would have provided. This would have mattered if I had been trying to use a DVI-I cable, but I wasn't. Both ends of my cable had the felicitous DVI-D Single Link male connector, which would be able to get along with almost any female aside from the DVI-A.
It seemed, then, that my HDMI cable, running from the computer to the monitor, was looking to be teamed up with a male DVI-D adapter, not a male DVI-I. (As a mnemonic, I thought of the harder-to-mate DVI-I male as Idiosyncratic, in contrast to the more agreeable Dude.) The male DVI-D could be either Single or Dual Link, thouigh the former would be more suitable for other possible connections in the future. On the other hand, not being familiar with video electronics, I thought maybe the greater number of pins in the Dual Link might yield more pleasing results.
I viewed some debates on DVI vs. HDMI. The sense I got was that HDMI was more convenient (easier connectors, audio inline) but DVI-D was more reliably good. If my monitors had offered an HDMI jack, I could have tried an HDMI Y-adapter to get around the single HDMI jack on the computer, but that wasn't in the cards for me now.
I was able to order a generic HDMI cable and adapter for about $7 total. When they arrived, I hooked them up. The HDMI-connected secondary monitor looked great! Now I had a new problem. The primary monitor, connected by DVI cable, was blank. Device Manager (in Control Panel) was fine with the idea that I now had only one monitor. Device Manager > View > Show Hidden Devices did not make any difference. I went into Device Manager > Display Adapters > ATI Radeon HD 4250 > right-click > Properties > Driver tab > Update Driver > Search automatically. It said I already had the best driver. I ran a search. There were the inevitable suggestions to update drivers, but that didn't seem to be helping these people.
I tried unplugging the HDMI. Now the DVI was working. Replugged the HDMI; now no DVI. Evidently I could have one or the other but not both. Tried using VGA instead of DVI for the primary monitor. Same result: I could have one monitor or the other, but not both. But then, correction: unplugging and replugging did give me VGA and HDMI working simultaneously: primary VGA, secondary HDMI. No VGA ghosting at the moment. Funny, Device Manager listed both monitors as being Analog. Maybe I didn't need HDMI -- maybe I should have just tried using VGA (primary) - DVI (secondary) instead of the opposite, DVI-VGA?
Anyway, I left it like that: VGA primary, HDMI secondary. When I finished this writeup for posting, a couple weeks later, there was still no ghosting, so this seemed to be a fix.