Note than $n$ is a parameter for the functions.
For some constants $c_1, c_2$ and $n_0,$c_1n^2\le an^2 + bn + c \le c_2n^2 for all n > $n_0.
Consider any quadratic function f(n) =an^2 + bn^2 + c$, where $a, b$, and $c$ are constants and $a > 0$. Throwing away the lower-order terms and ignoring the constant yields $f(n) = \Theta(n^2)$. Formally, to show the same thing, we take the constants $c_1 = a/4$, $c_2 = 7a/4$ and $n_0 = 2 \cdot\max(\lvert b\rvert / a, \sqrt{\lvert c\rvert/a}).
Q1: How were c_1, c_2, n_0 derived? They seem to be arbitrary. The book does not provide any kind of explanation on how they are derived.
For some constant c, n_0$ $0\le cn \le n^2$ for all $n > n_0
This can be seen by taking c = a+ \lvert b \rvert$ and $n_0 = \max(-1, -b / a)
Q2: How was c, n_0$ derived? Again they seem to be arbitrary and lack an explanation.