  The main purpose of this article is concerned with the existence and the
precise forms of the transcendental solutions of several refined versions of
Fermat-type functional equations with polynomial coefficients in several
complex variables by utilizing the Nevanlinna theory of meromorphic functions
in several complex variables. In fact, we investigate the existence and forms
of the transcendental solutions of non-linear quadratic trinomial equations in
$\mathbb{C}^n$. As a consequence of our result, we show that solutions of
binomial equations in $\mathbb{C}^n$ can be explored and this exploration
broaden the scope of the study of functional equations in $ \mathbb{C}^n $. The
results we obtained are improvements over certain recent findings, noted as
remarks. In addition, some examples relevant to the content of the paper have
been exhibited in support of the validation of each results. Further, we
discuss situations when solutions of such equations does not exist.
