Thank you very much for answering! I was starting to fear no one would have a comment.
A "google-ing" of the problem reveals that a lot of people have this problem.
THE PROBLEM IS A LITTLE DIFFERENT NOW, in that I did a lot of work since I described the problem.
( I uninstalled IE 8, then I backed out of XP SP3, this then allowed me to uninstall IE7, and I ended up with IE6, running XP SP2
I then worked my way back forward:
I installed XP SP3
I then SKIPPED IE7,
and installed IE8 right away, on top of IE 6, bypassing ( skipping ) IE7
So , I am left with a system: running XP SP3, with IE 8.
AND,WHERE I AM AT IS, THE PROBLEM MANIFESTS AS FOLLOWS:
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I start an Internet Explorer v. 8, ... the browser window opens, and TWO iexplore.exe processes run in taskmgr
I then "x" out of the browser window ... one of the iexplore.exe processes go away, but ONE of them remains in taskmgr, as a "phantom" process.
OKAY, NOW THAT I'VE DESCRIBED THE CURRENT MANIFESTATION, LET ME GIVE YOU DETAILS:
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I cannot tell you when it *started*. I *can* tell you that I noticed it about a week ago.
I am the IT tech support person for a team of 10 people or so. SO, this is not my computer. I was given this PC to work on, because it was "very slow" (which it was).
But as I tore away at it, I took care of most of the problems, with the one problem remaining being:
When I start Internet Explorer ( version 8 ), TWO iexplore.exe processes start running. I know that this is NORMAL.
But when I "x" out of the browser, ONE of the iexplore.exe processes keeps on running, does not go away.
So, you are in the situation where;
there is no browser window; but there is ONE iexplore.exe process running.
Then, I bring up Internet Explorer again, while it is running there are now THREE iexplore.exe processes running.
I "x" out of the browser, and one of the iexplore.exe processes go away, but now there are TWO iexplore.exe processes running, though no browser window.
Bring up Internet Explorer again, ... there will now be FOUR iexplore.exe processes running
( the TWO PHANTOM ones, and the TWO that are "NORMAL" for the browser window that is actually running.
But "x" out of that broser, and, again, I ended up with THREE phantom iexplore.exe processes running,
and so on, and so on.
In other words, every time I start an Internet Explore, and then "x" out of it, I end up with a phantom iexplore.exe process after I close the browser window.
ON THIS MACHINE, the fellow did not have Firefox, only Internet Explorer.
( I happen to prefer Firefox, and, as a WORK-AROUND, I installed Firefox on my colleague's PC, and told him to use Firefox, avoid Internet Explorer.
But that is no answer, really. I'd like to figure out why these phantom iexplore.ext processes hang around, because:
(a) he might prefer IE over Firefox
(

he might accidentally start IE rather than Firefox
© as you know, some Microsoft sites, e.g., Windows Update, is not friendly to Firefox at all, wants IE
(d) IT IS POSSIBLE, is it not, that these "phantom" iexplore.exe processes are doing some malware / spyware / whatever work
(e) and even if it is perfectly innocent, not a malware situation,
each of these phantom iexplore.exe processes eat up memory, the more you create, the more memory they use,
SO , OF COURSE, I suspect that at least PART of the "very slow" performance problem had to do with these phantom iexplore.exe processes runnin gin the backgroun.
AGAIN, THANKS VERY MUCH FOR RESPONDING.
I DO HOPE YOU CAN HELP.