Mercenaries 2: World in Flames

You can do this one of a few ways:

1) Do missions for them, should unlock after a few.
2) Bribe them by talking to Fiona at the PMC
3) Join a co-op game where the host player already has them at maximum mood

All the friendship achievements can be earned quite easily.
1. When you finish the introduction to a fraction, bribe them.
2. Capture 1 hvt or complete 1 simple mission.
that should make the fraction happy enough.
Or,
1. join a co-op game with a friend that has the fraction a full mood already.
Hope this helps!