  We study the inflationary phenomenology of a non-minimally coupled Einstein
Gauss-Bonnet gravity theory, in the presence of a scalar potential, under the
condition that the gravitational wave speed of the primordial gravitational
waves is equal to unity, that is $c_T^2=1$, in natural units. The equations of
motion, which are derived directly from the gravitational action, form a system
of differential equations with respect to Hubble's parameter and the inflaton
field which are very complicated and cannot be solved analytically, even in the
minimal coupling case. In this paper, we present a variety of different
approximations which could be used, along with the constraint $c_T^2=1$, in
order to produce an inflationary phenomenology compatible with recent
observations. All the different approaches are able to lead to viable results
if the model coupling functions obey simple relations, however, different
approaches contain different approximations which must be obeyed during the
first horizon crossing, in order for the model to be rendered correct. Models
which may lead to a non-viable phenomenology are presented as well in order to
understand better the inner framework of this theory. Furthermore, since the
velocity of the gravitational waves is set equal to $c_T^2=1$, as stated by the
striking event of GW170817 recently, the non-minimal coupling function, the
Gauss-Bonnet scalar coupling and the scalar potential are related to each
other. Here, we shall assume no particular form of the scalar potential and we
choose freely the scalar functions coupled to the Ricci scalar and the
Gauss-Bonnet invariant. Certain models are also studied in order to assess the
phenomenological validity of the theory, but we need to note that all
approximations must hold true in order for a particular model to be valid.
