I've been trying to find the coefficient of $x^{80}$ as you can see below. I'm having trouble following the steps here, specifically the second line. How did it go from the first line to the second line? What happens after that? I'm lost after the first one. Haven't really seen anything like this before.
Note that $1-x^8=(1-x^4)(1+x^4)=(1-x)(1+x+\dots + x^7)$, so we have the second line.
Since it is a generating function, $\frac{1}{(1-x^8)^3} = 1+a_8 x^8 +a_{16}x^16 + a_{24} x^24 +\dots$, or equivalently the $i$-th coefficient of the series is $0$ unless $8$ divides $i$.
So if we want to count the $80$-th coefficient of $(1+\dots +x^8 + \dots x^11)(1+a_8 x^8 +a_{16}x^16 + a_24 x^24 +\dots)$, we only have to pay attention to terms $1$ and $x^8$, since they are the only terms that might give $x^80$ at the end.