FOIL Calculator

Multiply two binomials using the FOIL method (First, Outer, Inner, Last) with step-by-step solutions.

Enter Binomial Terms

(1x + 2)(3x + 4)

First binomial: (ax + b)

Second binomial: (cx + d)

Result

Expanded Expression
3x² + 10x + 8
F First 1x · 3x = 3x²
O Outer 1x · 4 = 4x
I Inner 2 · 3x = 6x
L Last 2 · 4 = 8

Step-by-Step Solution

(ax+b)(cx+d) = acx² + (ad+bc)x + bd

What Is the FOIL Method?

The FOIL method is a technique for multiplying two binomials. FOIL is an acronym that stands for First, Outer, Inner, Last, which describes the order in which you multiply the terms. It is one of the most commonly taught methods in algebra for expanding the product of two binomial expressions.

Given two binomials (a + b) and (c + d), the FOIL method produces:

(a + b)(c + d) = ac + ad + bc + bd

How FOIL Works

First

Multiply the first terms of each binomial together.

(a + b)(c + d) -> a · c

Outer

Multiply the outer (first and last) terms of the expression.

(a + b)(c + d) -> a · d

Inner

Multiply the inner (second and third) terms of the expression.

(a + b)(c + d) -> b · c

Last

Multiply the last terms of each binomial together.

(a + b)(c + d) -> b · d

Combining Like Terms

After applying the FOIL method, the Outer and Inner products often contain like terms (both are first-degree terms when multiplying linear binomials). These like terms should be combined to simplify the final expression. For example, when multiplying (2x + 3)(x + 5):

  • First: 2x · x = 2x²
  • Outer: 2x · 5 = 10x
  • Inner: 3 · x = 3x
  • Last: 3 · 5 = 15

Combining like terms (10x + 3x): 2x² + 13x + 15

When to Use FOIL

  • Expanding binomial products in algebra classes and homework.
  • Simplifying expressions before solving equations.
  • Factoring quadratics (FOIL in reverse).
  • Working with polynomial multiplication where both factors are binomials.

Special Products

Certain binomial products follow recognizable patterns:

  • Perfect Square: (a + b)² = a² + 2ab + b²
  • Difference of Squares: (a + b)(a - b) = a² - b²
  • Sum and Difference: These patterns are direct consequences of the FOIL method and are worth memorizing for speed.