Can the background image of a div overflow the div without causing scroll bars?

I am coding my design portfolio site, and I have a header and footer image that I need to extend horizontally beyond the div containers. I can't figure out how to this, or how to keep horizontal scroll bars from appearing as the window sizing decreases.

MORE INFO: I'm creating this site to be responsive. The background image I want to appear is 1038px wide, but my footer container can only be 768px wide for the vertical tablet version, because I don't want the scroll bars to appear. I want to be able to set the background image to the footer div for ease of image placement without the div cutting off the image.

Thanks!