How can one make up for past days or months of fasting if they do not remember whether they had fasted or not, and is the making up (qada’) required to be accompanied by feeding (it’am), and is it permissible to feed the questioner's family?

If the doubt about fasting days after puberty is merely a waswasah (whispering from Satan), then it must be disregarded. However, if it is not waswasah, then the original principle is that one's responsibility is occupied with fasting, and it is obligatory to make up the days about which there is doubt regarding their fasting, based on the Prophet's (peace be upon him) saying: "Allah's debt is more worthy of being fulfilled." Expiation (Kaffarah) is not required except for breaking the fast by sexual intercourse. There is no ransom (Fidyah) for delay due to ignorance of the ruling. Some scholars hold the view that the original principle is the acquittal of responsibility, so what the mukallaf (accountable person) doubts does not obligate him except with certainty. However, the more cautious approach is to make up the days about which there is doubt. It is not permissible to give Kaffarah or Fidyah to those whose maintenance is obligatory upon oneself. It is permissible to give it to poor relatives whose maintenance is not obligatory upon oneself.