What is the characteristic equation of $a_{n+2}+2a_{n}=0$? So if we let $a_{n}=Cr^n$ then we have $Cr^{n+2}+2Cr^n=Cr^n(r^2+2)$. So I got that the characteristic equation $r^2+2=0$ but it should be $r^2+2r=0$ apparently. How is that?

The characteristic equation of :
$$a_{n + 2} + \alpha a_{n + 1} + \beta a_n = 0$$
is :
$$r^2 + \alpha r + \beta = 0$$
In your case $\alpha = 0$ and $\beta =2$.
