Me again, problem fixed. I went to the Dell Support Center website and found this: sprtcmd.exe not responding message

It basically said: SPRTCMD.EXE is a file used by Dell Support Center. The error occurs if Microsoft Windows XP does not have the Microsoft .NET Framework Version 2.0 installed. The .NET Framework 2.0 must be installed to prevent the error from reoccurring.

So I installed Microsoft.net Framework Version 2.0, shutdown, restarted, shutdown, problem fixed.

The curious thing is that I do NOT have version 2.0 on my desktop, and do NOT have this problem on my desktop. It only happened on the laptop. Can't remember which is older, but apparently the support software is different on the two.

So it wasn't a virus or trojan and I am now again doing restore points, although I only have yesterday and today so far, having wiped it out during my selective startup last night. Thanks to all for your help, which ended up steering me down the path to a solution.

If Microsoft and Dell both knew about this issue, why wasn't it included in a Microsoft or Dell automatic update... (why ask why).