делать всё возможное

do one's best; put one's best foot forward; set best leg foremost; leave no stone unturned; bend over backwards; go out of one's way to (You're right... strategically smart... but, you see, here's why I don't like you and I intend to go out of my way to never hire you.); go all out (- The boss wants us to go all out.); lean over backward; bend over backward; fall over backward; push; take great pains (its practitioners take great pains to emphasise that what they do is different from …); give 110%; give a hundred and ten percent; give it one hundred and ten percent; fall over oneself; put best leg foremost; be doing everything it can (CNN Alex_Odeychuk); make every attempt (TranslationHelp); do everything possible; do everything in one’s power; do one’s utmost; go to great lengths (Anglophile); be on one's mettle

Игорь Миг

do one's part; make every effort

amer.

snap oneself in two (Taras)

austral., slang

flat out like a lizard drinking

idiom.

work every angle (Even close allies keep things from one another – and work every angle to find out what's being held back. Даже близкие союзники скрывают кое-что друг от друга – и делают все возможное, чтобы выяснить, что скрывают от них. VLZ_58)

inf.

do one's damnest (do one's best dj_formalin)

Makarov.

do one's best; move heaven and earth; put one's best foot forward

do everything we can (to + inf.; New York Times Alex_Odeychuk)

rhetor.

try his best