When I teach students to multiply binomials, I teach the standard FOIL method because that is the method taught in every text book that I've ever used. I teach three other methods to my students so they have a choice. Not everyone finds the FOIL easy to do, especially if you end up multiplying a trinomial by a binomial.
The cool thing about this particular method is that the reverse is the grouping process for factoring so there is a connection between this method and factoring trinomials.
The second alternative method is what I call the regular multiplication method. I open by asking how you multiply 27 x 32. After they guide me through the process to the answer, I show them how to multiply the binomials using the same format. I draw the dotted lines in so they see how it works and I tell them to include the plus or minus so the correct signs allow the correct answer.
Many of my students find this method so easy to do because it follows the same procedure they use for integers and for them it is comfortable to make the transference.
I admit, I had to learn the lattice method because its not one I'd encountered before but if its what my students already know, I decided to apply it to binomials. It really does work. If you've never seen it before the carry part is in the same square but in a different diagonal.
If anyone is interested, I could create a short video showing you how to do the lattice method of multiplication or as it is applied to binomials, let me know and I'll do it.
The students find having four choices a bit difficult because they are so used to being told to do it "This Way!" but they like finding the method that works best for them.