How do I Correct for False Positive using MBRContains in a mySQL db?

Question

I've got a mySQL database. I'm testing points to see whether they are in particular polygons. Unfortunately, I'm getting false positives. So, for example, if I test to see if different points are in polygon ABCD, the MBRContains query returns 6 points, when I know for a fact that only 5 of the points are in that polygon.

I know that MBRContains uses minimum bounding rectangles instead of the actual polygon, so it's not always accurate. Is there a second check that I can make to check whether the point is a false positive?

I know that mySQL isn't the best with spatial queries. I've read that the PostGIS extension for PostgreSQL is better. But, my set-up is mySQL.

How do you deal with false positives? Any advice?