I have heard that Gauss studied integer binary quadratic forms with the middle term divisible by two $$f(x,y) = a x^2 + 2bxy + c y^2. $$ However, more modern treatment does not do this. Buell's "Binary Quadratic Forms" starts with the line:
We consider here the binary quadratic forms in two variables $f(x,y) = a x^2 + bxy + cy^2$ of discriminant $b^2 - 4ac = \Delta$.
But Gauss must have handled the odd terms somehow, otherwise there would be no forms with a discriminant equal to (+/-) an odd prime. And when it comes time to discuss composition of quadratic forms, I'd think it would cause havoc with missing elements in the set of forms of a given discriminant.
Similarly, in discussions of more modern work, the binary quadratic form $$f(x,y) = a x^2 + 2bxy + c y^2$$ and binary cubic form $$f(x,y) = a x^3 + 3bx^2y + 3cxy^2 + d y^3$$ seem to be important. For example, in Bhargava's "Higher composition laws I", Bhargava cubes can be related to cubic forms if the middle terms are a multiple of 3, and to pairs of the quadratic forms if the middle terms are even.
So what is special about the "form" of these forms?
Does this continue, and binary quadratic forms are more convenient to study if they look like this?
$$f(x,y) = a x^4 + 4b x^3 y + 6c x^2 y^2 + 4d x y^3 + e y^4$$
And, more explicitly, in the case of the binary quadratic forms:
What advantage did an even middle term have that made Gauss study it that way?
And how does the modern study (where we do allow odd middle terms) relate to his work?
Does this also apply for cubic and higher binary forms?
Apologies for the soup of questions, but these extra factors of Binomial coefficients are confusing me. Especially when it sounds like there is some simple reason why these are equivalent ways of studying these objects. Unfortunately, this simple reason is not obvious to me.
(For reference, I am a curious undergraduate, probably reading stuff before I should ... so no, I do not understand everything in Buell's book or Bhargava's paper, but I'd very much like to learn more.)
Note: This could be viewed as an extension of this question:
What's so special about the form $ax^2+2bxy+cy^2$?
Although the answers seems focused on the generality part of the question rather than the issue about the coefficient being "even".