Just bought the HP 640, has a brand new cartridge,...

SOURCE: HP 640 Fax...can't recieve faxes

Try disconnectiong the handset completely

SOURCE: Not Receiving Faxes

It sounds like you're not set for AUTO RECEIVE. Try unplugging power cord and re-plug in

SOURCE: My HP 640 will not print. Ran the test, no print.

If these are new cartridges, make sure you removed all of the protective plastic wrap that comes when packaged.  usually its red , yellow or orange plastic tab that should be removed.
If these are not new cartridges, you may need to try another set. 
Try running the head cleaning utility under the printer utilities options HP is very good at having this utility available to trouble shoot most common problems.  If you don't have this utility, try updating your printer driver. (basically uninstall the printer and download the driver from the HP website, and reinstall the printer using the latest driver).
Hope this helps..
~enov8

SOURCE: HP 640 Fax series not printing

If it is a color printer, try printing just grayscale or black& white.  It is is already set to print just b/w, try printing in color.  Basically you may be out of ink.  Run the printer utility to clean the heads and see if anything changes.  A new set of cartridges should fix the problem, I believe.
Hope this helps...
~enov8

SOURCE: Problem with my hp 640 fax

once you dial the number you fax will automatically start sending the the Fax signal which sounds like short beeps. This is done so the receivers phone (when answered) will recognized the beeps and switch to fax receive mode.  If you are getting the recording that you mentioned above, that means the number you dialed was not successful in making the connection.  Once you dial the number you should hear the phone ringing followed by some churping noises when the other end answers the call... then other churping and gargle like sounds initially as the fax proceeds.  
sounds like your are having trouble with the number you are dialing.
Hope this helps..
~enov8