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.