This is a Heyawacky puzzle. The rules are the same as Heyawake, but non-rectangular regions are allowed.
Rules of Heyawake:
Shade some cells of the grid.
Shaded cells cannot be orthogonally adjacent; unshaded cells must be orthogonally connected.
There cannot be a horizontal or vertical line of unshaded cells that passes through two borders.
If a number is in a room, there must be exactly that many shaded cells in that room.
(The reason this genre is given a different name is the third rule: in regular Heyawake, it is often phrased as "line of unshaded cells that is in at least three rooms" rather than "that passes through two borders". That phrasing is equivalent if all rooms are rectangular, but not necessarily if there are nonrectangular rooms.)