You already know the total will be 416, and that the bookshelf costs $80, so you already have a part of the equation. The rest of the equation can be 4b, where b is equal to books. So, you'd write it like this:
4b+80=416
To solve this, you have to move 80 to the other side. To do this, subtract 80 from both sides so that you get this:
4b=336
Then, divide both sides by 4 to get the maximum amount of books Mr. Harrison can buy:
b=84
Mr. Harrison can buy a maximum of 84 books, along with the $80 bookshelf.