Paint-stained concrete

First, let's look at what you've already tried, starting from the well-known fact that oil and water don't mix.

Soap and water wouldn't remove oil-base paint even if it was still wet. Ajax and bleach work well on water-soluble things, not so well on oil-based products. There are paint removers, and then there are paint removers. Some work well; many don't work worth a damn. Chances are pretty good that the People's Republic of Kalifornia doesn't permit the sale of any of the ones that actually work. Look at the label of the paint remover you tried. If it does not contain methylene chloride, it isn't going to work. Get one that does. Examples include ZipStrip and StripFast.

Tried a pressure washer?

If that fails, sandblast it. A house in my neighborhood changed hands recently. The new owner decided he didn't like the look of painted brick (who can blame him?) and hired a crew to sandblast it. It looks beautiful. You can't tell that it was ever painted (at least from the street -- I haven't inspected it up close).

Oil-based paint, obviously. :-)

Roller. How long the nap needs to be would depend on how smooth the concrete is; unless it's really rough, 3/8" is probably enough, maybe 1/2".

If you're subject to sub-freezing temperatures, I don't think I'd paint that. Moisture's going to get into that slab, no matter what you do. I'm not sure I'd want a layer of paint over the top -- too much chance of it spalling when the temperature drops below freezing. Your best bet IMHO is to try to remove the paint: ZipStrip first, then pressure washer, then sandblasting.

And put down dropcloths next time you paint. :-)

When you can't beat 'em, join 'em :o) My neighbor slopped dark brown wood stain all over the concrete atrium floor in our condo. It is quite old, and blotchy with stains, but the dark brown really stood out. I took out my acryllic hobby paint, a stencil brush, and painted back in the concrete colors - mostly gray, white, little umber, little yellow, touch of blue. I painted in thinly a little darker color, for the "pores" of the concrete, and then very sparingly a lighter tone for the texture of the concrete. Not too bad, if I say so. Much better than dark brown blotches, and you can't tell it's painted.

PS: Muriatic is not a stain remover. It is a concrete remover.

Since nothing else would remove the paint stain, I thought muriatic acid would etch the concrete far enough to get through the paint stain.

I doubt acid does well in removing things that are oil based. As others have suggested, try a product specifically made to remove paint.

If you decide to paint it, there are lots of paints specifically for concrete. I recently used one of the concrete stain products from HD or Lowes and was very happy with it. They call it a stain, but it was fairly solid, covering more like a paint. I don't remember which one it was specifically, but the color was New England Slate, which might narrow down the brand. Prep on your concrete should be no more than power washing it and caulking any cracks.