Representation of integers by positive definite ternary quadratic polynomials with linear terms There is a huge amount of research dealing with analysis of representation of integers by a quadratic polynomials only with terms of degree 2 (here is for example review of some known methods by J.Hanke: http://www.math.ubc.ca/~cass/siegel/hanke-ternary.pdf). There is a theory where we can expess corresponding theta series as a sum of Eisenstein series and cusp forms and then make some estimation on representation number as coefficients of theese series. 
My question is the next: is it possible to extend these methods for analysis of representation of integers by quadratic quadratic polynomial with linear terms, especially when $n=3$ (for example, $m=5a^2+5b^2+5c^2+2a+2b+4c$)? My knowledge of theory of Modular Forms is not so good, so I would ask you an advice. Thank you in advance. 
 A: Yes, it is possible to extend these methods, although the picture is somewhat 
less clear than in the quadratic form case. When one discusses representations by a quadratic polynomial, this is equivalent to asking for representations by a coset of a lattice. 
The theta series of a lattice coset is still a modular form, although usually for a smaller group than $\Gamma_{0}(N)$. This is described well in Iwaniec's book "Topics in Classical Automorphic Forms".
There is also a notion of genus for a lattice coset (see this paper of Wai Kiu Chan and Byeong-Kweon Oh), and this presumably says something about the Eisenstein series part of that theta series, although I don't think there's an analogue that's been worked out of the beautiful formula of Siegel that represents the Eisenstein series coefficient as a product of local densities. (At least, this is the impression I got from Wai Kiu Chan when I talked to him about it.) 
Note that Chan has had a few students work on some problems about ternary quadratic polynomials (although not much from the modular forms perspective yet); in particular take a look at the work of Anna Haensch (see this paper and this paper) and James Ricci.
In some cases, the problem really is as simple as completing the square. (Exercise: Prove that every non-negative integer can be written in the form $a^{2} + b^{2} + c^{2} + ab + ac + bc + a + b + c$.)
