Have a good reason for asking your professor for an extension. You don't want to ask a professor for an extension because you have not been planning ahead or managing your time well or because you have been partying. No professor will take those requests seriously. However, if you have a health problem (like the flu), learning disability, death in the family or some other good reason, it is likely that your professor may grant the extension.
Collect any proof, if available, of the reason why you need the extension. For example, if you have had a death in the family, your professor may want to see an obituary. If you have a learning disability, a note from your doctor or tutor may be warranted. Bring these items with you when you ask your professor for the extension.
Know how long of an extension you will need. Try not to ask for too much time. If you ask for weeks, a professor will likely not be on board with your request. You could ask for several hours or several days. But, be sure to have a time frame in mind so you can specifically ask the professor for what you need.
Set up a time to meet with your professor to discuss your request. The professor's office hours may be a good time to meet. Let the professor know that you will be coming to the office hours to discuss an issue and the time that you will arrive. Don't wait until the last minute to meet and request the extension. Talk with your professor as soon as you know there will be a problem with the deadline for your assignment.
Practice and prepare what you want to say to your professor. Go over in your head or out loud what you will need to say, making sure that you cover all of your points. If you need to make notes for yourself, that is ok too.
Take a deep breath and meet with your professor. Explain why you need the extension, show the proof (if necessary) and suggest to the professor how much time you need. No matter the outcome, it never hurts to ask. Also, remember to always thank your professor for their time in meeting with you and for considering your request.