What is (5+3i)*(3+5i)

The idea here is to explain to the student that i^2=-1, and to expand and simplify the brackets in question just like they would with simple algebra. Therefore:
(5+3i)*(3+5i)=15+25i+9i+15i^2=15+34i-15=34i
Students can often multiply out the brackets correctly, but then they are not sure what to do with term involving "i^2". Confusion is even bigger if there are minus signs floating around.