The important thing to remember here is that the slime on the floor will make your men lose battle fitness, the yellow number by the little figure. When that number reaches 2 or less the man will panic. This is normally not too big of a problem except that the mother alien keeps shooting at targets. If you guy does not move, he will probably get shot.
Obviously you need to keep all your guy's moving, the pallets on the floor are safe so get onto those. Always leave at least one turn to move. The mother will only target one guy at a time but, it is slow to target so just move one square and you'll keep ahead of it.
The big limbs of the mother can be "shot off", but it take a little while with most weapons. I usually only shoot the near one off, and leave the other.
Once you get to the clear land the spider start pouring out of the complex in the back to be ready with a double fre machine gun and maybe some double fire guns. Something that will not overheat, or something that kills many with one shot like a mine-thrower.
You will earn a commendation from this mission.
You can buy THESE weapons after the mission.