Can the results about quadratic forms be generalized to arbitary polynomials of degree $2$? I came across two theorems dealing with the question whether a quadratic form can represent every positive integer (Details here : https://en.wikipedia.org/wiki/15_and_290_theorems )
But there are several restrictions. If I understand it correctly, the theorem can only be applied for homogenous polynomials of degree $2$. Moreover, the quadratic form must be positive definite.

Can the theorems be generalized to arbitary polynmials of degree $2$ ?

In other words, can we determine, whether an equation like $$x^2-xy+y^2-2yz+2z^2-3x+5y+6z-8=n$$  has an integer solution for all positive integers $n$ ?
I do not require to find all solutions or to decide whether there are solutions at all. It would be sufficient to find an $n$ for which there is no solution or to show that there is a solution for every $n$.
 A: This is one of the many questions in number theory which has a simple statement, but potentially a very difficult answer. At present it's not clear how to generalize the analytic machinery that makes things like the 290-theorem possible to prove. In particular, the theory of modular forms plays a key role here, and while it is still true that there is a modular form attached to a quadratic polynomial, there is no particularly nice formula for the contribution of the Eisenstein series.
I don't know off-hand a quadratic polynomial for which it's unknown whether it represents every positive integer, but it is quite difficult for Anna Haensch and Ben Kane to prove (in this paper) that every sufficiently large positive integer is a sum of three $m$-gonal numbers if $m \not\equiv 2 \pmod{3}$ and $4 \nmid m$.
There are ternary quadratic forms whose representation behavior we do not yet understand. In particular, it is not known whether every odd number can be written in the form $x^{2} + xy + 5y^{2} + 2z^{2}$ (although I proved in this paper that it does, assuming the generalized Riemann hypothesis).
