Part of my under carriage keeps dropping. ???

It's very difficult to tell what part it is from that picture, and angle. Please take a wider angle shot further away from the car, so we can see where it is located relative to other things.

Also, take a picture from underneath the car.

 

unfortunately, the dealer sounds clueless. they should look at it again to see why it's not holding. maybe you need a new panel, speed bumps and potholes shouldn't make it fall down.
i hope they are doing this for free, since the initial charge.
is that the oil change flap? some people just remove it.

 

The correct fasteners would probably cure the problem.

 

I recognize that fastener type, but as @xliderider says: your picture is so close it's hard to understand. Maybe even more description as to where you're seeing this. Maybe one of the middle under panels, along the side of the car??

And as @bisco says: the dealership is not doing the repair right, so far. Maybe now they will. I'm suspecting the portion of the fastening system attached to the car body is torn or missing.

I think someone had a similar issue, a few months back, and those fasteners were not cheap. There might be something you can cobble yourself, if you're willing to get down there and have a look. Unfortunately, it's expensive and frustrating to have Toyota mechanics dealing with fussy plastic bits.

 

If the fasteners were replaced with the correct ones and the panel still comes loose, then there is probably damage to the hole that the fastener is supposed to hold onto. Most likely enlarged or stripped out or torn so that the fastener cannot grip properly. If that is the case, a nylon or aluminum washer with an appropriate hole size can be epoxied to the panel so that the fastener has something to grip.

 

Items 24 and 25 here?

FLOOR & RAILS for 2010 Toyota Prius | Toyota Parts

Again, just going by the shape of the hanging down fastener, really no clue where the pic is, need wider field of view and/or more description.

And of course there's a couple of 24's and 25's, with identical part numbers.

If it comes down to having to order the whole panel in order to replace a fastener, I'd seriously consider getting under there yourself, really seeing what needs doing. It's just flimsy plastic, and very likely something can be cobbled that will make it just as secure as stock.

 

And.......The correct people.....Toyota should be absolutely embarrassed when their "Certified" mechanics can't properly reattach a plastic underbelly panel .....

 

You can at least describe where in the car it is. For example, ahead of driver's side front wheel, passenger side, etc.

 

Could be those parts 24/25. It seems like the fasteners are sold as part of the panel, which is frustrating.