unfortunately I don't quite get how I should reach Step 2 with the Distributive Laws and I'm getting also confused how it is allowed to put the OR's in Step 1 in brackets.
The result I'm getting in Step 1 looks like this:
(¬R ⋁ (S ⋁ (¬Q ⋀ P ))) ⋀ ((¬S ⋀ (Q ⋁ ¬P)) ⋁ R)
Appreciate any help, thanks.
The problem is that the right hand side of the original formula is $S \lor \neg Q \land P$, and it is not clear how to read that. Is that to be interpreted as:
$S \lor (\neg Q \land P)$ (this is how you interpreted it as)
or as:
$(S \lor \neg Q) \land P$ (this is what the answer key did)
These two way statements are not equivalent, and so that's why you go a different answer.
Now, because of this, most logic textbooks will simply not accept any statement of the form $P \lor Q \land R$, and force you to place parentheses to disambiguate. However, a few textbooks have some kind of priority scheme to disambiguate, so that's what may be happening here.