> Since I implemented the + operator, the += operator
> should have been created for me. (And it was, as I
> confirmed.)

+= is syntactically an operator, but it is not a method. a method does
not know which variable the object was originally in.

a += b

does exactly

a = a + b

(no matter what the type is, afaik)

it also means the left-hand side must be a variable name, not an object.