In rome total war how do you get factions to become your protectorates

1) offer them lots of money
2) conquer much of their empire then offer them 1 or 2 regions back for them becoming ur protectorate
3) siege their last city, blockade their port, destroy their army - they may ask to become protectorate, If you ask them they will probably accept if u give them money or good terms,
4) find a faction with no allies and many enemies - the Seleucids often are in this state - and ask them to be ur protectorate so you can 'protect' them

Any or several of these will do