  The goal is to make a global study of the family QsnSN of all real quadratic
polynomial differential systems which have a finite semi-elemental saddle-node
and an infinite saddle-node formed by the collision of two infinite singular
points. This family can be divided into three different subfamilies, all of
them with the finite saddle-node in the origin of the plane with the
eigenvectors on the axes and (A) with the infinite saddle-node in the
horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C)
with the infinite saddle-node in the bisector of the first and third quadrants.
These three subfamilies modulo the action of the affine group and time
homotheties are three-dimensional and we give their bifurcation diagram with
respect to a normal form, in the three-dimensional real space of the parameters
of these forms. In this paper we provide the complete study of the geometry of
the first two families, (A) and (B). The bifurcation diagram for the subfamily
(A) yields 29 phase portraits for systems in QsnSN(A) counting phase portraits
with and without limit cycles, while the bifurcation diagram for the subfamily
(B) yields 16 phase portraits for systems in QsnSN(B) under the same
conditions. Case (C) will yield quite more cases and will have an independent
paper in short. Algebraic invariants are used to construct the bifurcation set.
The phase portraits are represented on the Poincar\'e disk. The bifurcation set
of QsnSN(A) is not only algebraic due to the presence of a surface found
numerically. All points in this surface correspond to connections of
separatrices.
