It is also important to clarify what mathematical context you are working in.
For example, if a,b are numbers they commute and then (a+b)(a−b)=a2−b2 , but if they are square matrices then they may not commute so (a+b)(a−b)=a2−ab+ba−b2 is the best you can say without further information.
Another thing to note is that for some purposes (a+b)(a−b) may be the best form for the expression, for example you might be interested in knowing when this is zero given you know a , so you need to be aware of the real goal of what you are doing.
Answers & Comments
It is also important to clarify what mathematical context you are working in.
For example, if a,b are numbers they commute and then (a+b)(a−b)=a2−b2 , but if they are square matrices then they may not commute so (a+b)(a−b)=a2−ab+ba−b2 is the best you can say without further information.
Another thing to note is that for some purposes (a+b)(a−b) may be the best form for the expression, for example you might be interested in knowing when this is zero given you know a , so you need to be aware of the real goal of what you are doing.