Let $I,J $ ideals of $R$ is $(IJ)[X] = I[X]J[X]$ true?
Attempt: The inclusion $\supseteq$ is obvious. For the other inclusion, take $f \in (IJ)[X]$. Then we can write $f$ as a sum of polynomials of the form
$$ijX^k$$ where $k \geq 0, i \in I, j \in J$. But
$$ijX^k = i(jX^k) \in i J[X] \subseteq I[X]J[X]$$
and thus $f \in I[X]J[X]$ as well.
Is this correct? (just need a quick sanity check).
Yes, that is correct.
Also you may prove it in this way: $$I \ \text{ Is Right Ideal } \ \Longrightarrow I[X]J[X] \subseteq (IJ)[X]$$ $$J \ \text{ Is Left Ideal } \ \Longrightarrow I[X]J[X] \subseteq (IJ)[X]$$