Factorising Polynomial over the real number field

hey guys I was just wondering whether I'm doing this right

for example the question asks to factor $\displaystyle z^6+8$ over the real number field

Heres what I did

$\displaystyle z^6+8=(z^2)^3+2^3$

$\displaystyle =(z^2+2)(z^4-2z^2+4)$

but when I checked the answers they got

$\displaystyle z^6+8=(z^2+2)(z^2+\sqrt6z+2)(z^2-\sqrt6z+2)$

they got this by pairing up the roots

which is more factored than my expression, the factors I got are not factorised enough right?, I have tried difference of 2 squares, yet they don't give me the same as the answer as they are complex

$\displaystyle =(z^2+2)(z^4-2z^4+4)$

$\displaystyle =(z^2+2)((z^2+1)^2-3i^2)$

$\displaystyle =(z^2+2)(z^2+1-\sqrt3i)(z^2+1+\sqrt3i)$

- Is the method I used originally wrong, since it is not fully factored?
- How would I be able to factorise it to the right answer (eg. provided by the textbook) by just continuing with the factors I originally got, without using the pairing up of roots method the text book used. Is it possible?

Thanks

Aonin (Nod)