A polynomial in $\mathbb Z[x]$ not generated by $5$ and $x^2+2$.

Question

I want to show that the ideal of $\mathbb Z[x]$ generated by the polynomials $5$ and $f(x)=x^2+2$ is strictly included in $\mathbb Z[x]$ so I wanted to find a polynomial $P\in \mathbb Z[x]$ that cannot be generated by $5$ and $f(x)$. I took $P=X^2+1$ and then I supposed that there exist $Q(x)$ and $R(x)$ in $\mathbb Z[x]$ such that $X^2+1=5 Q(x)+ (x^2+2)R(x)$ then in $\mathbb Z_5[x]$ this would give $(x-2)(x-3)=(x^2+2)Q(x)$ which implies that $(x^2+2)Q(x)$ is reducible modulo 5 which is not true because $x^2+2$ is not reducible modulo 5. Hence $x^2+1$ cannot be generated by $5$ and $f(x)=x^2+2$ is this reasoning correct and is there any easier way to find such an example.