Is this a correct application of the distributive law to $ (\neg P \wedge \neg Q \wedge R) \vee (\neg P \wedge Q \wedge \neg R)$? Is this a correct application of the distributive law?
$$\begin{align}
(\neg P \wedge \neg Q \wedge R) \vee (\neg P \wedge Q \wedge \neg R) &\equiv \phantom{\wedge}(\neg P \vee \neg P) \wedge (\neg P \vee Q) \wedge (\neg P \vee \neg R) \\
&\phantom{\equiv}\wedge (\neg Q \vee \neg P) \wedge (\neg Q \vee Q) \wedge (\neg Q \vee \neg R) \\
&\phantom{\equiv}\wedge (\phantom{\neg}R \vee \neg P) \wedge (\phantom{\neg}R \vee Q) \wedge (\phantom{\neg}R \vee \neg R)
\end{align}$$
 A: Yes, you have correctly distributed those two conjunctions.
$${\phantom{\equiv~}(\neg P \wedge \neg Q \wedge R) \vee (\neg P \wedge Q \wedge \neg R)\\ 
\equiv\\ \phantom{\equiv~}{\phantom{\,\wedge\,}(\neg P \vee \neg P) \wedge (\neg P \vee Q) \wedge (\neg P \vee \neg R) \\\wedge (\neg Q \vee \neg P) \wedge (\neg Q \vee Q) \wedge (\neg Q \vee \neg R) \\\wedge (\phantom{\neg}R \vee \neg P) \wedge (\phantom{\neg}R \vee Q) \wedge (\phantom{\neg}R \vee \neg R)}}$$
However, there is another way to apply distribution: to distribute out the common factor.
$${\phantom{\equiv~}(\neg P \wedge \neg Q \wedge R) \vee (\neg P \wedge Q \wedge \neg R)\\ 
\equiv\\ \phantom{\equiv~}{\neg P\wedge ((\neg Q\wedge R)\vee(Q\wedge\neg R))}}$$
You can then apply your distribution technique to the right conjunct , and simplify the expression in a few more steps:
$${\phantom{\equiv~}\neg P\wedge ((\neg Q\wedge R)\vee(Q\wedge\neg R))\\\equiv\\ \phantom{\equiv~}{\neg P\wedge ((\neg Q\vee Q)\wedge(\neg Q\vee\neg R)\wedge (R\vee Q)\wedge(R\vee \neg R))}\\\equiv\\~~~\ddots}$$
