You didn't state your model of touchsmart, however, if you have the standard vga port out you could use it with either the M1 plus or limited. Up to the tx2 1244 lineup you have that port out for video but no composite. The newer models likely have HDMI out as well.
If you have the correct connector, then you will get a prompt from Windows 7 to ask if you want to use either display singly or both simultaneously. You can also adjust through your display settings the external projector capability. The models listed are close to the same as the laptop, 1280 x 800 is the laptop but if they support only 1280 x 720 that can be adjusted as above. I do that all the time on my HDTV to get 1920 x 1080. The cable if not included is a VGA 15-pin double male end. This is only video, if you want audio you will need a double male 1/8" (3.5mm)stereo cable, but probably this little projector will not provide sufficient sound out.

Hope this helps!

Hi Peter-
You are probably speaking of the new M1 ultimate version, which I quickly took a look at. According to the spec sheet, there are two pertinent items. I am assuming you have the vga out capability, otherwise you are stuck with composite. I would go with the vga if possible. The technique for putting out 1920x1024 works best with extending rather than mirroring the display and then using the display adjustments through control panel you tweak the second one up to the high resolution.
In this case, our second consideration is the limitation of the graphics processor on the M1. It states that it is native 800x600, so this is a standard definition resolution screen ration of 4:3. It does state that it is capable of a VGA compatible input of 1024x768. Since that number exceeds the native resolution, it will either downscale the input to the 800x600 or convert to an overscan (interlaced) format in order to put it in the frame. In any case, since this is lower than your current display, when Windows detects the device, it will likely adjust your main display as well. So mirroring is your better option, but your screen image will drop down. The pain here is that windows often forgets how to recover to your original resolution so it may need it done manually.

Hope this helps