Maximal ideal in the ring of polynomials over $\mathbb Z$ 
Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal.

I tried first see that $5$ and $x^2+2$ are both polynomial in that ring but how can i get that is maximal ? some help please.

Say you add some $f\notin\langle 5, x^2+2\rangle$, and consider the ideal $I'=I+(f)$. We want to show this is the entire ring $\mathbb{Z}[x]$.
Well, dividing by remainder, we can write $f=(x^2+2)g+h$ where $h$ is linear. Reduce $h$ mod $5$ and we write $h=ax+b$ where $0\le a, b\le 4$. By assumption that $f\notin I$, we get $h\neq 0$.
I believe there is some nice algebra to reduce the casework here, to show basically that using $h, 5$ and $x^2+2$, you can get $1$ and hence $I'=\mathbb Z[x]$, but I'm not sure off the top of my head how to prove it in a unified, clean way.
Also, if a reference would help you, in the first few pages of chapter 1, Reid Undergraduate Commutative Algebra, he proves that for any PID $A$, maximal ideals of $A[x]$ are exactly those of the form $(f,p)$ where $p\in A$ is prime and $f\in A[x]$ is irreducible mod $p$. Maybe running through that proof with your special case would be helpful?
