Why is this method called FOIL?

FOIL is an acronym for First, Outer, Inner, Last—the order in which you multiply pairs of terms from two binomials. It is a mnemonic device to help you remember all four products you must find, rather than forgetting the outer and inner terms.

Can I use FOIL with three binomials or trinomials?

FOIL applies specifically to two binomials, so no. For more terms, you apply the Distributive Property step by step: multiply the first factor by each term in the second, then combine those results. This is the general approach that FOIL is based on.

Is FOIL the only way to multiply two binomials?

No. The Distributive Property is the universal method. FOIL is just a shorthand memory trick for organizing the four products when you have two binomials. Some people prefer to draw a box (area model) to avoid missing any products, and that works just as well.

What is the connection between FOIL and the Quadratic Formula?

FOIL is used to expand binomials into quadratic expressions (polynomials of degree 2). Once you have a quadratic in standard form ax^2 + bx + c, you can use the Quadratic Formula to solve for the roots. Learning FOIL helps you recognize quadratic patterns and set up equations to solve.

FOIL Method

The FOIL Method expands products of two binomials by multiplying each term in the first binomial by each term in the second.

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

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What each symbol means

What FOIL Method takes

Symbol	Meaning
$a$	The first term of the first binomial; if confused with $b$ , you lose the binomial structure and cannot apply FOIL correctly.
$b$	The second term of the first binomial that is added to $a$ ; misplacing this term into the second binomial breaks the entire multiplication.
$c$	The first term of the second binomial; if its sign is misread (positive as negative or vice versa), all four products will have the wrong signs.
$d$	The second term of the second binomial; forgetting to multiply by $d$ leaves out two of the four products FOIL requires.

When to use it

Use FOIL when you need to multiply two binomials together, such as expanding $(x + 2)(x + 3)$ or solving quadratic equations.

Level

Usually taught in: Algebra I

Worked examples

1. Expand two binomials with small positive integers

Problem

Expand

(x + 2)(x + 3)

$F: x \cdot x = x^2$
First terms: multiply the first term of each binomial.
$O: x \cdot 3 = 3x$
Outer terms: multiply the leftmost term of the first binomial by the rightmost term of the second.
$I: 2 \cdot x = 2x$
Inner terms: multiply the rightmost term of the first binomial by the leftmost term of the second.
$L: 2 \cdot 3 = 6$
Last terms: multiply the last term of each binomial.
$x^2 + 3x + 2x + 6 = x^2 + 5x + 6$
Combine like terms: add the two middle terms $3x + 2x = 5x$ .

Answer: $x^2 + 5x + 6$

This is the simplest case of FOIL, where all terms are positive integers and it clearly shows how all four products appear before combining like terms. The final answer is a trinomial (three terms), which is typical after FOIL on two binomials.

2. Expand binomials with a negative term and a leading coefficient

Problem

Expand

(2x - 3)(x + 4)

$F: 2x \cdot x = 2x^2$
First terms: multiply the first term of each binomial, paying attention to the coefficient $2$ on the $x$ .
$O: 2x \cdot 4 = 8x$
Outer terms: the coefficient $2x$ times the $4$ from the second binomial.
$I: (-3) \cdot x = -3x$
Inner terms: the negative sign on the $-3$ carries through, giving a negative product.
$L: (-3) \cdot 4 = -12$
Last terms: multiply $-3$ by $4$ , keeping the negative sign.
$2x^2 + 8x + (-3x) + (-12) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$
Combine like terms: $8x - 3x = 5x$ .

Answer: $2x^2 + 5x - 12$

This example shows how to handle negative terms and coefficients other than $1$ in FOIL. Keeping track of signs is crucial: the negative term $-3$ affects each of its two products. After all four products are found, combining like terms gives the final answer.

3. Expand binomials in a real-world context

Problem

A basketball arena is building a rectangular VIP seating section. The length is

(x + 5)

meters and the width is

(x + 2)

meters. What is the area of the section?

$\text{Area} = (x + 5)(x + 2)$
Area of a rectangle is length times width, so we must expand this product.
$F: x \cdot x = x^2$
First terms: $x$ times $x$ .
$O: x \cdot 2 = 2x$
Outer terms: $x$ from the first binomial times $2$ from the second.
$I: 5 \cdot x = 5x$
Inner terms: $5$ from the first binomial times $x$ from the second.
$L: 5 \cdot 2 = 10$
Last terms: $5$ times $2$ .
$x^2 + 2x + 5x + 10 = x^2 + 7x + 10$
Combine like terms: $2x + 5x = 7x$ .

Answer: $x^2 + 7x + 10 \text{ square meters}$

Expanding binomials using FOIL is essential in real-world geometry and business problems where dimensions are expressed with variables. Once expanded, the area expression $x^2 + 7x + 10$ can be used to find the actual area by substituting a specific value for $x$ , or to solve for $x$ given a target area.

Common mistakes

Where FOIL Method usually goes wrong

The mistake	Why it is wrong	The fix
Writing $(a + b)(c + d) = ac + bd$ , ignoring the middle terms.	FOIL requires all four products: you cannot skip the outer and inner terms $ad$ and $bc$ .	Remember that both the first and second terms of one binomial must multiply both terms of the other. Write out all four products: $ac + ad + bc + bd$ , then combine like terms.
Writing $(x - 2)(x - 3) = x^2 - 6$ by incorrectly handling the negative signs.	Negative signs apply to each product separately: $(-2) \cdot (-3) = +6$ , not $-6$ , and you must include the inner and outer terms as well.	Carefully apply the sign to each product: $F: x^2$ , $O: -3x$ , $I: -2x$ , $L: +6$ . Then combine: $x^2 - 3x - 2x + 6 = x^2 - 5x + 6$ .
Writing $(2x + 3)(x - 1) = 2x^2 + 3x - 2 - 3$ , treating unlike terms as if they were alike.	Only true like terms can be combined; while $-2x$ and $3x$ are like terms, the constants $-2$ and $-3$ come from different products and cannot simply be added.	Perform all four steps of FOIL carefully: $F: 2x^2$ , $O: -2x$ , $I: 3x$ , $L: -3$ . Group the four products and combine only the like terms: $2x^2 + (-2x + 3x) - 3 = 2x^2 + x - 3$ .

Tips and when to use something else

FOIL is a memory aid for the Distributive Property applied twice: $(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd$ . If you forget FOIL, this is the method to fall back on.
FOIL only works for two binomials (four terms total). If you have $(a + b + c)(d + e)$ or any factor with three or more terms, use the Distributive Property step by step instead.
After FOIL, always look for like terms to combine—this simplifies your answer and is usually where signs are checked. Many mistakes come from skipping this step.
If you are expanding $(a + b)^2$ or $(a - b)^2$ , you can use FOIL, but the perfect square trinomial formula— $(a + b)^2 = a^2 + 2ab + b^2$ —is faster and less error-prone.

Related formulas

Distributive PropertyThe Distributive Property lets you multiply a number by a sum or difference by distributing the multiplication to each term separately.Combining Like TermsCombining like terms merges coefficients of identical variables to simplify expressions and solve equations more efficiently.Quadratic FormulaThe Quadratic Formula solves any equation of the form ax^2 + bx + c = 0 in one step, giving both solutions at once without factoring.Completing the SquareCompleting the square rewrites a quadratic as a perfect square plus a constant, making it easier to solve and find the parabola's vertex.Order of OperationsOrder of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.

Frequently asked questions

Why is this method called FOIL?: FOIL is an acronym for First, Outer, Inner, Last—the order in which you multiply pairs of terms from two binomials. It is a mnemonic device to help you remember all four products you must find, rather than forgetting the outer and inner terms.
Can I use FOIL with three binomials or trinomials?: FOIL applies specifically to two binomials, so no. For more terms, you apply the Distributive Property step by step: multiply the first factor by each term in the second, then combine those results. This is the general approach that FOIL is based on.
Is FOIL the only way to multiply two binomials?: No. The Distributive Property is the universal method. FOIL is just a shorthand memory trick for organizing the four products when you have two binomials. Some people prefer to draw a box (area model) to avoid missing any products, and that works just as well.
What is the connection between FOIL and the Quadratic Formula?: FOIL is used to expand binomials into quadratic expressions (polynomials of degree 2). Once you have a quadratic in standard form $ax^2 + bx + c$ , you can use the Quadratic Formula to solve for the roots. Learning FOIL helps you recognize quadratic patterns and set up equations to solve.

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Reviewed 2026-09-18