Order all the prime ideals of $\Bbb{R}[X]$ and list the maximal ideals.

Question

Let $R=\Bbb{R}[X]$. I need to do the following:

1. find all prime ideals
2. find all maximal ideals
3. show containments of prime ideals
4. indicate the minimal prime ideal
5. is $R$ an integral domain
6. is $R$ a local ring

My idea was the following:

Let me first remark that $R$ is a PID, so every ideal can be written as $(P)$ for $P\in R$. From here we immediately can deduce that $R$ is an integral domain.

We first want to find all the maximal ideals. We know that these are all ideals $(P)$ such that $P$ is irreducible in $R$. But this are all ideals of the form $$(aX^2+bX+c)$$ for $a,b,c\in \Bbb{R}$ such that for all $z\in \Bbb{R}$ $az^2+bz+c\neq 0$. From here we can also say that $R$ is not a local ring.

Now let us talk about the prime ideals. We clearly know that all maximal ideals are also prime ideals. So we only need to check if there are more ideals which are prime but not maximal. I claim that there aren't further prime ideals. Indeed let $Q$ be not a irreducible element of $R$. So $$Q=(X-a_1)...(X-a_n)$$ for $a_i\in \Bbb{R}$. Clearly $(X-a_1)...(X-a_n)\in (Q)$ but $(X-a_i)\notin (Q)$ since $\deg(X-a_i)=1<\deg(Q)$, hence $Q$ is not prime. Thus we have seen that all prime ideals are the maximal ones.

Now we see that $(0)$ is the minimal prime ideal and I think there is no maximal prime ideal.

Is this correct like this or am I completely wrong?