Is this simplification correct, and if so, what law does it illustrate?

Question

Can I simplify

$(F \land \neg M ) \lor (F \land A) \lor (F \land M \land G)$

to

$F \land (\neg M \lor A \lor (M \land G) )$

? And if so, what law does this illustrate?

Answer 1

$(\color{blue}{F} \land \neg M ) \lor (\color{blue}{F} \land A) \lor (\color{blue}{F} \land M \land G)$

We can use the distributive law to get:

$\color{blue}{F} \land (\neg M \lor A \lor (M \land G) )$ You can simplify by first using commutativity of disjunction to obtain $$F \land( A \lor\lnot M \lor(M\land G))$$ and then $$F\land (A \lor((\lnot M\lor M) \land (\lnot M \lor G))$$

Which can be simplified to

$$F\land (A \lor \lnot M \lor G))$$

Answer 2

The distributive law is the factoring law. You have factorised the expression by taking out the common factor of F.

So it is the distributive law.