Fill in the blank with the appropriate demonstrative. did you ask laura if she wanted some of sandwich before you threw it away?

The demonstrative would be 'that'  
Did you ask Laura if she wanted some  of __that__ sandwich before you threw it away.  It would be 'that' because it's asking about one things and referring to one thing in general not more than one: words sandwich and it hints the demonstrative.
 
But if the sentence said sandwiches and them instead then the demonstrative would be those. 

They were constantly being overlooked as a people and were being controlled by people who weren't really native to their homeland.