Factoring over $\mathbb{Z[x]}$, $\mathbb{Q[x]}$, $\mathbb{R[x]}$, $\mathbb{Z_3[x]}$, $\mathbb{Z_7[x]}$

Question

I want to factor $2x^5$ + $2x^3$ – $4x^2$ – $4$ in $\mathbb{Z[x]}$, $\mathbb{Q[x]}$, $\mathbb{R[x]}$, $\mathbb{Z_3[x]}$, $\mathbb{Z_7[x]}$.
I did it for $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, but I am not sure it is OK.
$\mathbb{Z[x]}$ = $2(x^2 + 1)(x^3 - 2)$
$\mathbb{Q[x]}$ = $2(x^2 + 1)(x^3 - 2)$. It is the same, since all of the polynomials there are irreducible.
$\mathbb{R[x]}$ =$-√2* √2 (x^2+1)(∛2-x)(x^2+∛2 x+2^{2/3})$.
Now, I am not sure, what to do with $\mathbb{Z_3[x]}$, $\mathbb{Z_7[x]}$ and if what I did is right.
Thanks for all the help.