MySQL version 5.6 added ST_CONTAINS() function which will do a true spatial search if a point is within a polygon. You will need to do the following:
1) ST_CONTAINS( polygon, point ) : the first parameter should be the column name in the table containing your polygon.
2) The second parameter is a POINT datatype constructed using the spatial function GeomFromText()
3) The MySQL query below will return the row (neighborhood) which contains the lat/lng point.
SELECT * FROM your_table where ST_Contains( your_polygon_column, GeomFromText( 'POINT( your_lat, your_lon )' ) );