The word FOIL is one of the most popular acronyms in math. It is an acronym because each letter stands for another word. Here is the breakdown.

F = first

O = outer

I = inner

L = last

The words first, outer, inner, and last all pertain to the terms in a binomial. The best way to demonstrate this concept is to go over a few examples.

Example 1: Multiply the binomials using the FOIL method.

$$\left( {x + 3} \right)\left( {x + 4} \right)$$

If we examine the binomials above, the first term of each binomial is both \(x\). The outer terms are \(x\) and \(4\). The inner terms are \(3\) and \(x\). And finally, the last terms are \(3\) and \(4\). We just need to multiply them, then combine like terms.

$$F \to \left( x \right)\left( x \right) = {x^2}$$

$$O \to \left( x \right)\left( 4 \right) = 4x$$

$$I \to \left( 3 \right)\left( x \right) = 3x$$

$$L \to \left( 3 \right)\left( 4 \right) = 12$$

Now, we write all terms horizontally.

$${x^2} + 4x + 3x + 12$$

Finally, combine the like terms which are \(4x\) and \(3x\).

$${x^2} + 7x + 12$$

Example 2: Multiply the binomials using the FOIL method.

$$\left( {y + 2} \right)\left( {y - 6} \right)$$

Let's do an example where one of the binomial has a minus sign in the middle.

$$F \to \left( y \right)\left( y \right) = {y^2}$$

$$O \to \left( y \right)\left( { - 6} \right) = - 6y$$

$$I \to \left( 2 \right)\left( y \right) = 2y$$

$$L \to \left( 2 \right)\left( { - 6} \right) = - 12$$

Putting them together, we have

$${y^2} - 6y + 2y - 12$$

Then combing like terms, we get

$${y^2} - 4y - 12$$

Example 3: Multiply the binomials using the FOIL method.

$$\left( {k - 1} \right)\left( {k - 2} \right)$$

Let's try an example where both of the binomials have a negative sign in the middle.

Multiplying the first (F) terms:

$$\left( k \right)\left( k \right) = {k^2}$$

Multiplying the outer (O) terms:

$$\left( k \right)\left( { - 2} \right) = - 2k$$

Multiplying the inner (I) terms:

$$\left( { - 1} \right)\left( k \right) = - k$$

Multiplying the last (L) terms:

$$\left( { - 1} \right)\left( { - 2} \right) = 2$$

Putting all the products together, we obtain

$${k^2} - 2k - k + 2$$

Finally, combine the similar terms \(-2k\) and \(-k\). We know that \( - 2k - k = - 3k\).

$${k^2} - 3k + 2$$