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What’s wrong with FOIL?

The FOIL (First, Outer, Inner, Last) method to algebraic multiplication states that:

{$(a + b)(c + d) \quad = \quad \underbrace{a c}_{\text{First}} \quad + \quad \underbrace{a d}_{\text{Outer}} \quad + \quad \underbrace{b c}_{\text{Inner}} \quad + \quad \underbrace{b d}_{\text{Last}}$}

This is a special case of the Distributive Law. The FOIL method is very inefficient when multiplying lots of terms together, like in {$(a+b+f+g)(c+d+e)$}. Using FOIL, one gets

{$\Big(\underline{a+b}\,+\,\underline{f+g}\Big) \quad\quad \Big(\underline{c+d}\,+\,\underline{e}\Big)$}

{$= \underbrace{(a+b) (c+d)}_{\text{First}} \;+\; \underbrace{(a+b) e}_{\text{Outer}} \;+\; \underbrace{(f+g) (c+d)}_{\text{Inner}} \;+\; \underbrace{(f+g) e}_{\text{Last}}$}

{$= \; \ldots$}

and FOIL must be used again on each term. Instead it is better to think directly in terms of the Distributive Law. This becomes more obvious when we look at multiplication geometrically.

Beyond FOIL

The area of a rectangle of height {$a$} and width {$c$}

       c
   XXXXXXXXXX
 a XXXXXXXXXX
   XXXXXXXXXX

is {$a \times c$}. Similarly, a rectangle of height {$(a+b)$} and width {$(c+d)$} has area

          c  +  d
     XXXXXXXXXXxxx
  a  XXXXXXXXXXxxx
  +  XXXXXXXXXXxxx
  b  OOOOOOOOOOooo
     OOOOOOOOOOooo

is {$(a+b)\times(c+d)$}. This rectangle decomposes into four separate rectangles with areas {$a c, \, a d, \, b c, \, b d$}

         c       d                   c           d
     XXXXXXXXXX xxx              XXXXXXXXXX     xxx
  a  XXXXXXXXXX xxx           a  XXXXXXXXXX   a xxx
     XXXXXXXXXX xxx    ===>      XXXXXXXXXX     xxx

  b  OOOOOOOOOO ooo                  c           d
     OOOOOOOOOO ooo           b  OOOOOOOOOO   b ooo
                                 OOOOOOOOOO     ooo

demonstrating that the total area {$(a + b)(c + d)$} does indeed equal {$a c + a d + b c + b d$} as predicted by FOIL.

Now we can expand {$(a+b+f+g)(c+d+e)$} just as easily

          c  +  d + e
     XXXXXXXXXXxxxeeee
  a  XXXXXXXXXXxxxeeee
  +  XXXXXXXXXXxxxeeee
  b  OOOOOOOOOOoooEEEE
  +  OOOOOOOOOOoooEEEE
  f  CCCCCCCCCCcccBBBB
  +  DDDDDDDDDDdddGGGG
  g  DDDDDDDDDDdddGGGG
     DDDDDDDDDDdddGGGG

by noting the smaller rectangles that form the larger one:

                                            c          d       e
          c      d   e                 XXXXXXXXXX     xxx     eeee
     XXXXXXXXXX xxx eeee             a XXXXXXXXXX   a xxx   a eeee
  a  XXXXXXXXXX xxx eeee               XXXXXXXXXX     xxx     eeee
     XXXXXXXXXX xxx eeee
                                            c          d       e
  b  OOOOOOOOOO ooo EEEE             b OOOOOOOOOO   b ooo   b EEEE
     OOOOOOOOOO ooo EEEE    ===>       OOOOOOOOOO     ooo     EEEE

  f  CCCCCCCCCC ccc BBBB                    c          d       e
                                     f CCCCCCCCCC   f ccc   f BBBB
     DDDDDDDDDD ddd GGGG
  g  DDDDDDDDDD ddd GGGG                    c          d       e
     DDDDDDDDDD ddd GGGG               DDDDDDDDDD     ddd     GGGG
                                     g DDDDDDDDDD   g ddd   g GGGG
                                       DDDDDDDDDD     ddd     GGGG

Going row by row on the figure at the right, we can see that {$(a+b+f+g)(c+d+e) = a c + a d + a e \quad + \quad b c + b d + b e \quad + \quad f c + f d + f e \quad + \quad g c + g d + g e$}.

See the pattern? Every variable on the left side (i.e. {$a, b, f, g$}) is multiplied by every term on the right side (i.e. {$c, d, e$}), and all these products are added together. This is the Distributive Law in its most general form.