This question long-time bothering me.
Let $f(x) = 2x^2 + 5x+1$
Find the order of $\mathbb{Z_{10}}[x] / \langle 2x^2 + 5x +1 \rangle $
My friends said it is not difficult. He suggested the solution like the below
$\mathbb{Z_{10}}[x] / \langle 2x^2 + 5x +1 \rangle \simeq \mathbb{Z_{2}}[x] / \langle 2x^2 + 5x +1 \rangle \times \mathbb{Z_{5}}[x] / \langle 2x^2 + 5x +1 \rangle $
Question) Is it really true? then why? I'm doubt about his solution. Any help would be appreciated.
p.s.) Why having I the doubt for his solution.
The link I asked)Find the isomorphic ring and its order with $\mathbb{Z_{15}}[x] / \langle 3x^2 + 5x \rangle $
The link2 I asked) Product of the ideal and normal groups(Is this solution right?)
If $R$ is a unital commutative ring and $ab=0\in R,au+bv=1\in R$ then $$R\cong R/(b)\times R/(a)$$ through $$t \to (t\bmod (b),t\bmod (a)), \qquad (x\bmod (b),y\bmod (a))\to xau+ybv$$ Here $a=2,b=5$