Its difficult to evaluate the cause of gum infection
after multiple tooth extraction.
I would suggest you to get a thorough clinical as well as radiological examination.
The remaining root pieces has to be removed if present.
Take complete course of antibiotics-analgesics too.
Maintain good oral hygiene
Please discontinue the use of dentures until the socket heals completely.
Rinse frequently using warm saline and betadine
Bone filing has to be done if there is any bony protrusions.
I wish you good health.