  Polygons are described as almost-convex if their perimeter differs from the
perimeter of their minimum bounding rectangle by twice their `concavity index',
$m$. Such polygons are called \emph{$m$-convex} polygons and are characterised
by having up to $m$ indentations in the side. We use a `divide and conquer'
approach, factorising 2-convex polygons by extending a line along the base of
its indents. We then use the inclusion-exclusion principle, the Hadamard
product and extensions to known methods to derive the generating functions for
each case.
