Should I quit my job at the university bookstore? (employment, applying, interviewed)

[Aishata]

So essentially here is the situation. I am currently in a state university and during the previous summer I desperately needed a job. So I applied everywhere, until one store finally replied back, and it was the bookstore affiliated with the university (that provides the textbooks and stuff). When I interviewed in the position, they made it clear that they expected me to work past the summer into the fall. I accepted because I was desperate for money, despite the fact that I was already involved in an engineering club and was going to have another job in the fall.
Now, I have two jobs, school, and an engineering club. I began slacking off really bad at the university book store job and got severely reprimanded recently (was studying during the job and apparently not following procedures). I now want to put in my two weeks in two weeks. The problem is, I am applying for internships very soon for mechanical engineering, and I'm worried about the effect this would have on me. Obviously, I'm leaving this job off the resume (if I do quit) as I have been there for only 4 months. But would companies like Intel or Boeing do detailed enough background checks to find out about this job, which would likely give an unfavorable reference? (It was a tech salesman position selling cables, hardware, computers and stuff...)
Thanks so much for any advice or thoughts.
Note: I'm am also prepared for the possibility of being fired right after giving two weeks notice.

[fishbrains]

If you leave it off your resume, future potential employers will never know about it. Even if they did, they wouldn't care.

[tnff]

Frankly it wouldn't matter much to me if it was or wasn't on your resume. When I'm looking at freshly graduated engineers, I'd rather see internships, co-ops, research, etc related to the field you plan to enter. The reason is basically every new graduate has the same courses, and the all worked at the typical school jobs -- fast food, wait tables, store clerk. So none of these distinguish one from the other. Change the name and college and all the resumes are the same. It's also good that you're a member of the local engineering society. That's another distinguisher since many students don't join/participate. It helps set you apart.

[joe from dayton]

Unless you get fired for theft or something egregious, no one cares about your student employment in the book store. Your studies and internship opportunities should come first.

[Aishata]

Thanks for all of your replies!