4...b5 in the Slav

I would play 4 a4 in anticipation of that ...b5 idea.  rdecredico is right, this is like a bad QGA by black.

I don't see what the problem is, white is more than fine in that position.

You can easily recover your pawn and destroy blacks'.

Maybe this game can give you some idea:

https://www.youtube.com/watch?v=4F4bM5t2u7E&list=SPC8BC58A6FAA17F2D&index=3&feature=plpp_video

Commented by IM David Pruess from chess.com. It's not about the slav defense, but about a game where the black plays the variation that you mention here.

^ no. In the game you linked white has the N on c3. This makes a world of difference because black in that case can push his b pawn and gain a tempo. In that case black's pawn become a pain in the butt, in fact I play it as black.

In the position of the OP the N is on his home square, so white can regain his pawn easily.

I was thrown off by Black's confidence in playing these moves. I won eventually, but felt I made a meal of it. I was also thrown by the fact that Black had two connected passed pawns, with one already on b4.