Yes. Two triangle touching corner-to-corner. There would be six sides, but only 5 vertices because one vertex is shared.
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a shape with 7 vertices
A shape with four sides has more vertices because there are more points where edges (straight lines) meet than a 3-sided shape.
A 4 sided shape has 4 vertices whereas a 3 sided shape has 3 vertices.
any closed shape with 4 sides or more sides. The question does not specify only four vertices!
The number of vertices does not determine the number of faces. If the shape with 6 vertices was a quadrilateral based bipyramid, it would have 8 faces. A hexagonal based pyramid has 7 vertices and 7 faces. So more vertices does not necessarily imply more faces.