Microeconomics: collusion & cartel policy (answers included! just

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Hi, I have four rather long tutorial questions that I just do not understand. I’ve attached the questions along with the actual step-by-step answers given to us, but the answers make no sense. I don’t understand how the lecturer jumps from step one to step two, for example. Like, the first thing he might write is the completed formula with numbers plugged in and I’m like “what, where’d you get those numbers from? What’s happening?”. 

If someone could add in extra details, and explain the process like I’m five, along with the original formulas being used or what they’re called (or anything that could help me with what I should be studying a bit more) or just notes scribbled in like “he got these numbers here by… he expanded this out before getting the partial derivatives…” that’d be a great help.

Thanks for any takers!