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. 



Representation of integers by positive definite ternary quadratic polynomials with linear terms
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](https://arxiv.org/abs/1208.6060)), 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](https://arxiv.org/abs/1402.1640) and [this paper](https://arxiv.org/abs/1501.01654)) and [James Ricci](https://arxiv.org/abs/1505.00281).


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 $a^{2} + b^{2} + c^{2} + ab + ac + bc + a + b + c$.)


