Can I combine making up Ramadan fasts with fasting the six days of Shawwal?

According to the majority of scholars, yes, you can combine between the makeup fasts of Ramadan and the six days of Shawwal. You could have your primary intention being a makeup fast of Ramadan, but also another intention of it being a fast of Shawwal. The Prophet ﷺ said, 'Whoever fasts Ramadan and then the six days of Shawwal, it is as if he fasted the entire year.' That's in the Hanafi Madhab, Shafi'i Madhab, Maliki Madhab. It is permissible. It is preferable if you were to avoid combining. However, if you were to combine, then yes, it would be valid. In the Hanbali Madhab specifically, it is not permissible. You need to make up the Ramadan days first and then do the six days of Shawwal. That is specific for the Hanbali Madhab. But as mentioned, according to the majority of scholars, it is permissible to combine both intentions for that day. You could even have a third intention if you are fasting on a Monday or Thursday.