Proof of FOIL Modern Algebra

Question

I am trying to work through Birkhoff's A Survey of Modern Algebra independently, but am having difficulty getting off the ground with the proofs based on laws, rules, etc. I come from mostly soft analysis, and this is foreign. One problem is to prove (a+b)(c+d)=(ac+bc)+(ad+bd), using the properties of a commutative ring. Could someone please walk me through the proof, and how to do these in general?

Answer 1

$(a+b)(c+d)=a(c+d)+b(c+d)$ by distributive law.

$a(c+d)+b(c+d)=(c+d)a+(c+d)b$ by commutativity of multiplication.

$(c+d)a+(c+d)b=(ca+da)+(cb+db)$ by distributive law twice.

$(ca+da)+(cb+db)=(ac+bc)+(ad+bd)$ by associativity and commutativity of addition.

Answer 2

Hint: Start from the RHS and try to manipulate it into the LHS. For each bracketed pair, factor something out. This factoring is an application of distributivity. After doing this, apply distributivity a third time, and we're done.