this is from Seth Warner's Classical Modern Algebra:
Let $K$ be a field whose characteristic is not $2$, let $f=x^4+ax^2+b$ be an irreducible polynomial over $K$, and let $L$ be a splitting field of $f$ over $K$. show:
1) $[L:K]$ is either $4$ or $8$ [hint: consider $g=x^2 +ax+b$]
I did not use the hint in my answer. What I did was factor $f$ into $(x-c_1)(x-c_2)(x+c_1)(x+c_2)$ where $c_1$ and $c_2$ are in terms of $a$ and $b$. I then concluded that if $c_1$ is in $K(c_2)$, $[L:K]=4$. Otherwise $[L:K]=8$ since you will need adjoin roots twice and $4\times2=8$. Is this argument valid?
2) If $K$ is a totally ordered field and $b<0$, then $[L:K]=8$