So, I understand that multiplying ordinals has a distributive law on the left such that:
$\alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma$
What I am struggling with is if $\alpha$ is also another set of brackets, or even if there are three sets. So for example, how would you work out:
$(\omega\cdot2+1)(\omega\cdot3+3)(\omega\cdot4+7)$
Simply apply it repeatedly: for example, $(w\cdot 2 + 1)$ is an ordinal, so you have $(w\cdot 2+1)(w\cdot 3 + 3) = ((w\cdot 2 + 1)\cdot (w\cdot 3) + (w\cdot 2+1)\cdot3)$ and this is again an ordinal, so we have $(w\cdot 2 + 1)(w\cdot3 + 3)(w\cdot 4+7) = ((w\cdot 2 + 1)\cdot (w\cdot 3) + (w\cdot 2+1)\cdot3)\cdot (w\cdot 4) + ((w\cdot 2 + 1)\cdot (w\cdot 3) + (w\cdot 2+1)\cdot3)\cdot 7$.
You can then simplify that using the other properties of ordinal arithmetic (it would be far more efficient to do this as you go along).