This complex polyhedron can be considered to be the mutual Stott expansion either of x_p-4-o₂-4-o₂ by its dual, or the other way round. Accordingly the new vertex count is just the product of the former vertex count with the facet (in here: edge) count of its vertex figure. (This clearly will be independent of the chosen starting figure.) Faces here are both the faces of either starting polyhedron (shifted a bit out each) plus the mere prism product of the respective edges. The count of the formers clearly gets maintained in this expansion. The count for the additional facet type equates to the count of either of the former's edges.

In order to derive the respective edge counts, one observes that the respective edge counts per vertex are also the corresponding vertex counts of the vertex figure. Thus these are the vertex counts of the polytope according to that subsymbol of the current one, when both currently ringed nodes are omitted and then either the first (for the first edge type) or the last one (for the second edge type) gets ringed. (Within the current case of a polyhedron this simply amounts just in the index of the medial node for both cases.) Or, stated differently, its vertex figure happens to be (up to some scalings) just the lace prism x₂ || x₂.

Incidence matrix according to Dynkin symbol

xp-4-o2-4-x2

.    .    .  | 4p2N |   2    2 |  1   2   1
------------+------+----------+-----------
xp   .    .  |    p | 8pN    * |  1   1   0
.    .    x2 |    2 |   * 4p2N |  0   1   1
------------+------+----------+-----------
xp-4-o2   .  ♦   p2 |  2p    0 | 4N   *   *
xp   .    x2 ♦   2p |   2    p |  * 4pN   *
.    o2-4-x2 |    4 |   0    4 |  *   * p2N