Consider $f(x)=x^2+x+2$ over $Z_3$ and $F=Z_3[x]/<x^2+x+2>$. We can easily show that $f(x)$ splits in $F$. But How do we prove that $F$ is a splitting field?
Hint: It is mentioned that because $F$ is a two-dimensional vector space (basis can be easily seen as (x,1)) over $Z_3$, we know that $F$ is also a splitting field of $f(x)$ over $Z_3$.
By the quadratic formula, if $x_1$ is a root of $ax^2+bx+c$, then $-\frac{b}{a}-x_1$ is a root as well. So, as soon as you have an extension that contains a root, it contains the other one too.