STRONG recommendation - don't buy into school problems. Leave school issues at school. If the school keeps telling you, say, "Thank you for keeping me informed. What do you intend to do about it?" Make it clear that education is between the school and difficult child. You are the parent, not the teacher.
Because, as you have just discovered, when you get conned into taking on board the school's headaches, you are getting nowhere; in fact, you end up in a fruitless, exhausting screaming match. difficult child expects it; he hunches his shoulders ahead of it; you both get nowhere. Resentment build up on both sides.
Look at what easy child is doing. He's sending difficult child a clear message - "I love you. I care about you. I want you to do well. I will help you, to the best of my ability. But this has to be you, nobody else can force you to do this." He gave difficult child a pencil, so he could do his work. He was trying to make it easier to get started. Nothing said. No recriminations. Just, "Here's a tool. Now you can use it, if you choose."
There can be many reasons why a kid won't work at school. A kid in suspension can be feeling resentful for all sorts of reasons. If it's a kid who has trouble anyway, getting started with work, they are going to be less motivated than usual. If the school is letting him advance anyway, then what's the point of even trying? He may feel so out of his depth with some of the subjects, that he just doesn't know where to start. But he needs to feel he can approach you and ask for help, rather than see you always taking the school's side.
Basically, he gets this at school, home needs to be a refuge so he can regain his energy and feel more confident next day.
If the worst comes to the worst and he fails, and keeps failing, what adult education options are open to him in a few years time? We have a system here where people can go, who have not succeeded in getting the proper qualifications in school. They can do the courses as adults and often do much better because they're treated better and are motivated.
I do understand the frustration, though.
Marg