F + E(1-F) + (1-F)(1-E) = F + E - EF + 1 - E - F + EF
E(1 - F) = E - EF
by the distributive property
(1-F)(1-E) = 1 - E - F + EF
by using the method you described with a,b,c,d, which is just applying the distributive property twice. (We called it FOIL: Firsts, Outers, Inners, Lasts in Grade School)