Shit happens. You misread the syllabus. You had a (real actual) death in the family. Your boss tells you she needs you to work on your day off. Your kid is sick with the flu and now so are you. There are a litany of legitimate reasons why you would miss a deadline. Personally, I am extremely lenient with deadlines, going so far as to explain to my students that, apart from the end of the semester, I effectively don't enforce them. Obviously, not every professor is this way.
Regardless, your professor is a person capable of empathy and leniency to varying degrees. Your professor may have specific reasons why they have the policies they do, but often with the right approach they are willing to make exceptions. I have written this guide to help students feel confident in approaching their professor for an extension. This method is by no means fool-proof, but following these tips will make your professor more likely to grant you a reasonable extension.
1. Understand that your professor is human
Your professor is a person who decided to pursue a career in education because they want to share the knowledge they have with people who don't have that knowledge in the hopes that their students apply that knowledge to make the world a better place. With very few exceptions, professors want their students to pass their classes by learning the material and demonstrating their knowledge on the assessments they assign.
That said, your professor is also a person who has a life outside of teaching. Your professor likely has a love life, a family life, hobbies, social clubs, friends, etc. to which she wants to devote her time. When a professor determines the deadlines for assignments at the beginning of the semester, they are also planning to block out time in their schedule to devote to grading those assignments. If the professor is able to receive all the assignments at once, it means that they can grade all of them at once. When they get an assignment outside of that deadline, they are going to have to devote time that they planned to use for literally anything else to grade your assignment.
Therefore, you should begin by apologizing and acknowledging that this is inconvenient to the professor. You don't have to layer this on thick. A simple sentence will do. But the acknowledgement that you respect your professor as a person with a life rather than as an authoritarian grading machine will make them all the more likely to reciprocate.
2. Visit your professor's office hours
At many colleges and universities, professors are required to hold a particular number of office hours per week per number of credit hours they teach. Some professors utilize this time to complete their research, but regardless, the solitude can be eminently boring. We almost never have students visit office hours and would kill to have a student engaged enough in the course material to want to discuss it with us outside of class.
If you are able, you should try to arrange discussing the extension in person during office hours rather than sending a long apologetic email. In addition to providing us company, it demonstrates to the professor that you are willing to take time out of your day to visit them at a convenient time that they have already devoted to seeing students outside of class.
3. Contact your professor as soon as possible
As soon as it looks like you're going to miss the deadline, you should reach out to your professor to ask for an extension. This will allow the professor more time to rearrange his schedule so as to accommodate your request. It's pretty obvious, but I can't count the number of times I've had students wait until after the deadline to tell me about a mishap that happened before the deadline. The earlier notice you can give your professor, the more likely they will be able to rearrange their schedule to accommodate your extension.
4. Demonstrate you are doing the work
Many professors (not me) are immediately skeptical of any extension request. Perhaps rightfully so. I have given many extensions where the student still doesn't end up making good on the assignment. Depending on the assignment, this can be detrimental to the class (for instance, if graded assignments from other students function as answer keys for yours). You should demonstrate in some way that you are doing the work.
Ideally, this would mean bringing the completed assignment to office hours or attaching it to your extension request email. If you haven't had time to complete the assignment, it's a good idea to attach a rough draft, outline, notes, etc. to demonstrate that you have put in work and the work is simply not complete due to your circumstances. Even if you don't have deliverables, it's a good idea to summarize what work you have done and project what additional work you intend to do in order to complete the assignment.
5. Propose an alternative due date
What professors who say no to extensions are trying to avoid above all is being seen as not holding students accountable. Many professors (particularly women and professors of color who are often disrespected more than their white and male counterparts) are apprehensive as being seen as too soft and being walked all over by their students. For this reason, it is a good idea to propose to hold yourself accountable to a new due date within a reasonable time frame. "Reasonable" here should probably mean a week or less past the due date. If realistically, you will not be able to complete the assignment within a week (for example, if you're way behind on a long essay), propose deadlines for intermediate goals where you can deliver on concrete benchmarks toward completing the final project.
Like I said, these tips aren't fool-proof. Some professors absolutely do not budge on deadlines and many with good reason. It is important to remember that your professor is human and that grading assignments is often the most tedious part of their job. This is why professors often pass grading off to teaching assistants when they have the opportunity. However, if you follow these steps, you will communicate that you respect the professor, the job they are doing, and the knowledge they possess. Good luck with your homework.