I would like some feedback or verification if the steps I used below to calculate $(E:\Bbb Q)$ are correct. Thank you for your time.
Let $f(x)=(x^2+x+1)(x^2+x-1)$. If $E$ is the minimal splitting field of $f$ over $\Bbb Q$, I need to determine $(E:\Bbb Q)$.
First, I found the roots of $x^2+x+1$ to be $-\frac{1}{2}\pm\frac{\sqrt3}{2}i$, and the roots of $x^2+x-1$ to be $-\frac{1}{2}\pm\frac{\sqrt5}{2}$. I then concluded that both $x^2+x+1$ and $x^2+x-1$ are irreducible over $\Bbb Q$, and so $\Bbb Q(\sqrt3i,\sqrt5)$ is the minimal splitting field.
$\Bbb Q(\sqrt3i,\sqrt5)=\frac{\Bbb Q(\sqrt5)[x]}{(x^2+3)}$ and $\Bbb Q(\sqrt5)=\frac{\Bbb Q[x]}{(x^2-5)}$, so $(\Bbb Q(\sqrt3i,\sqrt5):\Bbb Q(\sqrt5))=2$ and $(\Bbb Q(\sqrt5):\Bbb Q)=2.$
Hence, $(\Bbb Q(\sqrt3i,\sqrt5):\Bbb Q)=(\Bbb Q(\sqrt3i,\sqrt5):\Bbb Q(\sqrt5))\cdot(\Bbb Q(\sqrt5):\Bbb Q)=2\cdot2=4$