Can you check if it is correct? My approach is as follows:
$R=\mathbb{Z}[x]/ (x^2+5)$ is isomorphic to $\{ax+b|a,b\in\mathbb{Z}\}$. We can consider two types of ideals in $R$. For $m\in\mathbb{Z}$ an ideal $(m)$ in R is $\{ax+b|a,b\in\mathbb{mZ}\}$. For $n\in\mathbb{Z}$, an ideal $(nx)$ is $\{cx+d|c\in\mathbb{nZ},d\in\mathbb{5nZ}\}$. Let $gcd(m,n)=i$ and $gcd(m,5n)=j$. Then, an ideal $(m,nx)$ in $R$ is $\{\alpha x+\beta|\alpha\in\mathbb{iZ},\beta\in\mathbb{jZ}\}$. However, the ideal $(m,nx)$ in $R$ can be expressed by one of two types of principal ideals.
I could not see where I was wrong. Can you help me?