What is the best way to approach 4yo son that behaves poorly and\or doesn't listen to you, ignores your words, and at the same time stay calm?
It seems the only thing he understands is a physical punishment or loud voice, anything else - just gets ignored.
I have already tried explaining and offering something in exchange. There was some result with the exchange idea, but if he really wants to do something (e.g. take away a toy from his small brother) - nothing is working.
When you say "explaining"... are you 100% positive he's paying attention? With my 3.5yo it makes a world of difference to walk across the room, pick him up and look him in the eye, nose to nose and run through the explanation. (and before anyone freaks out about man-handling him around, I'm talking the same kind of pick up you would do if he fell down and was crying for mommy/daddy... the point here is to get his attention fully on you.)
If he's squirming to get down and back to playing and not paying attention, I just tell him I need to talk to him, and I need him to pay attention, then he can go back to playing. Usually after repeating that once or twice he gets the hint and looks me in the eye and listens.
In a couple cases I've even had to take him out of the room/away from the distraction of whatever he was playing with to get him looking and listening to me. If even that can't calm him down long enough to listen to daddy... well... then playtime is over. Time to go do something else. (usually leave the play-space or park and go home, or it's nap time, amazing how that works out.)
The two main virtues of a parent of a young child, are patience and consistency. It's good to have simple rules your child can understand and be (very) consistent about them. If it does not behave there should be a consistent consequence to this behaviour. You should realise that it is good to explain why your child can not do something or should do something, but some things are just too dificult for young children to understand.
There are some good pointers to possible consequences in this thread: What are some suggestions for punishing a child?
Especially 'timeout' is something which is not only useful as a punishment for the child, but also for the parent to regain some calm and rationality. Bribing is a technique that you should be careful with, because it can encourage bad behaviour by rewarding it.
And lastly do not forget about positive enforcement. When your child (or its sibling) is behaving well, say so. Give compliments also at times, things are 'normal'.
I highly recommend the book Brain Rules for Baby. The author stresses empathy and labeling emotions: try to understand the child's feelings and tell them. For example, saying "I understand that you want to play with the toy and you don't want your brother to play with it. You are jealous."
+1 for positive reinforcement. If you don't want them to do something, don't forget to reward them for the absence of the behavior.
Also, +1 for consistency. Have a set of rules and do not relax on them. Giving a freedom and then having to take it away later will cause a fierce backlash.
It can actually be counter-productive to try to explain anything to a four year old during discipline.
In the example you give of taking the toy from his brother I am sure you have tried taking the toy, handing it back to the brother and stating simply, "Your brother had that first." in a calm fashion. I'm guessing the result was a tantrum.
My take on that one is let him have the tantrum but he has to have the tantrum in his room (or wherever is reasonably convenient) away from you and from brother. He can come back to play when he is done having a tantrum and ready to follow the rule that he may not take toys away from brother. No explanation necessary.
You'll have to deal with tantrums for a little while, but if you stay strong, don't give in, take a deep breath and enforce calmly he will get the point without you needing to force eye contact, lecture, or lose your own cool. If he needs to have a conversation with you where you explain the rule, be open to that after he is calm and ready to talk. This is the time to label his emotion as jealousy, to negotiate with "replacement toys" and to discuss turn-taking.
