I have sore gums behind the last lower tooth on left, very painful...using ibuprofen... can it be infection?

Could be: It sounds like you have a case of gingivitis. This can often worsening, if unchecked to periodontitis; see a dentist soon.

Maybe. See dentist : My first guess is trauma. Upper teeth extend just beyond lower teeth usually (even more so if there is an upper wisdom tooth present and none yet on lower). It is likely that some hard food was gnashed into gums, making them sore. Alternatively, if there is a partially erupted wisdom tooth, this is a great candidate for local infection (often accompanied by slight pain on swallowing or fever).

Pericoronitis: Pericoronitis is an inflammation of gum tissue surrounding a tooth, most often seen in the 3rd molar area where poor oral hygiene and partial eruption of the tooth causing inaccessibility to cleaning. It is painful with pain symptom comes and goes as the drainage establish. Extraction is the best option for the 3rd molars, especially when it is partially impacted.

Wisdom tooth: Likely an inflamed, infected wisdom tooth. See dentist or oral surgeon for assessment.