My 2004 ford taurus will not start on occasions after it has been driven for a while battery is good,alternator output is good does anybody have any suggestions:

Does the motor turn over, does it make a clicking sound?

If the starter motor won't always crank, read section T in the link below:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-honda-civics-46563.html

If it cranks but won't start, hook a scan tool up to the Data Link Connector and check for any codes being thrown. Next, I would replace the Motorcraft platinum spark plugs, if you have 50,000 miles on the vehicle. I own a 2000 Taurus, with the 3.0L Duratec engine, and know the 100,000 mile Ford recommended change interval is too ambitious. Performance starts to really drop after 50,000 miles on the plugs. Stick with the Motorcraft plugs that came in your car--don't go to cross-referenced plugs. Also, be sure and change the fuel filter every 25,000 to 30,000 miles--it's amazing what they filter out at times. Plugged fuel filters lower flow and pressure to the fuel rail, which can prevent cars from starting.

Be sure and change the transmission fluid and filter every 30,000 miles on your Taurus. Amsoil Universal Synthetic ATF or Mobil 1 Synthetic Multi-Vehicle ATF is definitely the way to go on Tauruses. If you don't take very good care of the transaxle, they can be problematic. These are very good cars.