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.

By the Correspondence Theorem, $(5,X^2+2)$ is maximal in ${\mathbb Z}[X] \Leftrightarrow (5,X^2+2)/(5)$ is maximal in ${\mathbb Z}[X]/(5)$.  Using an isomorphism, this is the same thing as saying that $(X^2 + 2)$ is maximal in $(\mathbb{Z}/(5))[X]$.  This ring is a PID, so we just need to show that $X^2 + 2$ is irreducible mod 5.  Since it has degree 2, this just means checking that it has no zeros mod 5.
