Multiplying Binomial - FOIL, Methods & Examples

TL;DR

Multiplying a binomial means distributing every term in the first bracket across every term in the second, so (a+b)(c+d)=ac+ad+bc+bd. This article walks through three methods (the distributive property, FOIL, and the box method), six worked examples, and the sign and like-term mistakes that trip students up most.

What Does It Mean To Multiply Two Binomials?

Multiplying two binomials means finding the product of two two-term expressions by multiplying each term of the first by each term of the second, then combining like terms. For (a+b)(c+d), that produces four products: ac, ad, bc, and bd. Add them, simplify, and you have the answer.

This is the same idea as the area of a rectangle whose sides are each split into two pieces. If one side measures a+b and the other measures c+d, the rectangle breaks into four smaller rectangles with areas ac, ad, bc, and bd. The total area is the sum. The algebra and the geometry are the same fact wearing different clothes.

The three methods, and when each one helps

The distributive property is the foundation. Distribute the first binomial across the second, then distribute again. It works for any polynomials, not just binomials, so it never lets you down.
FOIL is a memory aid for the distributive property applied to exactly two binomials. The letters stand for First, Outer, Inner, Last — the four products in the order you write them. FOIL is the quickest method for two binomials, but it only works for binomials.
The box method (also called the grid or area method) organises the four products in a 2-by-2 table. The grid that comes from the rusher forgetting a term is much harder to forget when every product has its own cell.

How Do You Multiply Binomials Using FOIL?

FOIL gives you the four products in a fixed order so none goes missing. Take (x+3)(x+5):

First — multiply the first terms: x⋅x=x²
Outer — multiply the outer terms: x⋅5=5x
Inner — multiply the inner terms: 3⋅x=3x
Last — multiply the last terms: 3⋅5=15

Now collect the four pieces and combine the like terms 5x and 3x:

x² + 5x + 3x + 15 = x² + 8x + 15.

The two middle terms almost always combine, which is why a multiplied pair of binomials so often lands as a three-term trinomial.

The special products worth recognising on sight

Square of a binomial: (a+b)² = a² + 2ab + b² and (a−b)² = a² - 2ab + b². The Outer and Inner products are identical here, so they combine into the single middle term 2ab.
Difference of squares: (a+b)(a−b) = a² - b². The Outer and Inner products cancel, so the middle term vanishes and you are left with just two terms.

Examples of Multiplying Binomial

Example 1

Multiply (x+2)(x+6).

First x⋅x=x².

Outer x⋅6=6x.

Inner 2⋅x=2x.

Last 2⋅6=12.

Combine the like terms:

x² + 6x + 2x + 12 = x² + 8x + 12.

Final answer: x² + 8x + 12.

Example 2

Multiply (x−4)(x+7).

First x⋅x=x².

Outer x⋅7=7x.

Inner (-4)⋅x=−4x.

Last (-4)⋅7=−28.

Combine the like terms:

x² + 7x - 4x - 28 = x² + 3x - 28.

Final answer: x² + 3x - 28.

Example 3

Multiply (2x+3)(x−5).

First 2x⋅x=2x².

Outer 2x⋅(−5)=−10x.

Inner 3⋅x=3x.

Last 3⋅(−5)=−15.

Combine the like terms:

2x² − 10x + 3x − 15 = 2x² − 7x - 15.

Final answer: 2x² − 7x - 15.

Example 4

Multiply (3a−2b)(4a+b).

First 3a⋅4a=12a².

Outer 3a⋅b=3ab.

Inner (−2b)⋅4a=−8ab.

Last (−2b)⋅b=−2b².

Combine the like terms:

12a² + 3ab - 8ab - 2b² = 12a² - 5ab - 2b².

Final answer: 12a² - 5ab - 2b².

Example 5

Multiply (x+12)(x−13).

First x⋅x=x².

Outer x⋅(−13)=−13x.

Inner 12⋅x=12x.

Last 12⋅(−13)=−156.

Combine the like terms:

x² + 12x - 13x - 156 = x² - x - 156.

Final answer: x² - x - 156.

Example 6

Multiply three binomials: (x+1)(x+2)(x+3).

Multiply two of them first:

(x+1)(x+2)=x² + 3x + 2.

Now multiply (x² + 3x + 2)(x+3):

x²⋅(x+3)=x³ + 3x².

3x⋅(x+3)=3x² + 9x.

2⋅(x+3)=2x + 6.

Add and combine like terms:

x³ + 3x² + 3x² + 9x + 2x + 6 = x³ + 6x² + 11x + 6.

Final answer: x³ + 6x² + 11x + 6.

Where The FOIL Shortcut Quietly Stops Working

The honest caution about FOIL: it is not a multiplication rule, it is a counting trick for one specific case. When students learn FOIL as the way to multiply, the method breaks the moment a problem has more than two terms.

Where Students Trip Up On Multiplying Binomial

Mistake 1: Dropping a sign on the inner or last product

Mistake 2: Forgetting to combine the two middle terms

Mistake 3: Multiplying only the first terms and the last terms

Practice Questions on Multiplying Binomial

Multiply (x+4)(x+6).
Multiply (x−5)(x+2).
Multiply (2x+1)(3x−4).
Multiply (a−3b)(2a+b).
Expand (x+7)(x−7).
Multiply (x+1)(x+2)(x−3).

Key Takeaways

The most common mistake is dropping a sign on a product or skipping the two cross-terms entirely.
This rule generalises to the binomial theorem and underlies polynomial multiplication of every size.