Math
Box Method Calculator
The box method, also called the area model, turns polynomial multiplication and factoring into filling in a grid.
To multiply, each term of one factor labels a row and each term of the other labels a column, and every cell is the product of its row and column. To factor a trinomial, the four terms go into a 2 × 2 box and the greatest common factor of each row and column gives the binomial factors. This calculator draws the box for both directions.
Each term becomes one row of the box.
Each term becomes one column of the box.
Try an example
Result
Result
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- Statement
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- Box used
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- ac-method pair (factoring)
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- GCF pulled out (factoring)
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- Degree
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- Check
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the box method calculator solves
Multiplying (2x + 3)(x - 4) means finding the area of a rectangle whose sides are 2x + 3 and x - 4. Cutting the rectangle into pieces by term gives a 2 × 2 box whose cells are 2x^2, -8x, 3x, and -12. Adding the cells gives 2x^2 - 5x - 12. The same box read backwards factors the trinomial.
Multiplying (2x + 3)(x - 4) with the box
| × | x | -4 |
|---|---|---|
| 2x | 2x^2 | -8x |
| 3 | 3x | -12 |
The two x terms sit on the off-diagonal; -8x + 3x = -5x. Product: 2x^2 - 5x - 12.
Factoring 6x^2 + 7x - 3 with the box
- ac = 6 × (-3) = -18. Find two numbers with product -18 and sum 7: 9 and -2.
- Put 6x^2 top-left, -3 bottom-right, and 9x and -2x in the other two cells.
- The largest number that divides evenly into every number in a set. of the top row (6x^2, 9x) is 3x; GCF of the bottom row (-2x, -3) is -1. GCF of the left column (6x^2, -2x) is 2x; of the right column (9x, -3) is 3.
- Read the edges: (3x - 1)(2x + 3).
How to use it
- Choose Multiply or Factor.
- For multiplying, type the two factors; the box can be any size, for example 3 × 2 for a trinomial times a binomial.
- For factoring, type the trinomial ax^2 + bx + c. Read the ac pair, the filled box, and the factors along the edges.
How to read the answer
In multiply mode the answer is the sum of every cell with like terms combined; cells on the same diagonal have the same degree. In factor mode the answer is the product of the row labels and the column labels; a GCF pulled out first appears in front, and the check line multiplies the factors back out.
Common mistakes and edge cases
- Forgetting to combine the diagonal cells before writing the product.
- Using a pair with the right product but the wrong sum in factor mode; both conditions must hold.
- Trying to factor a trinomial that is A whole number above 1 whose only factors are 1 and itself. over the integers, such as x^2 + x + 1. The calculator says so and points you to the quadratic formula.
- Not pulling out the GCF first: 4x^2 + 8x + 4 = 4(x + 1)^2.
Worked examples
Multiply two binomials
(2x + 3)(x - 4) = 2x^2 - 5x - 12
Result
2x^2 - 5x - 12
Binomial times trinomial (2 × 3 box)
(x + 2)(x^2 - x + 1) = x^3 + x^2 - x + 2
Result
x^3 + x^2 - x + 2
Factor with a leading coefficient
6x^2 + 7x - 3 = (3x - 1)(2x + 3)
Result
(3x - 1)(2x + 3)
Factor a monic trinomial
x^2 - 5x + 6 = (x - 2)(x - 3)
Result
(x - 2)(x - 3)
GCF first
4x^2 + 8x + 4 = 4(x + 1)(x + 1)
Result
4(x + 1)(x + 1)
Prime trinomial
x^2 + x + 1 does not factor over the integers
Result
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Frequently asked questions
What is the box method?+
An area model for polynomials. Terms label the rows and columns of a grid, and each cell holds the product of its row and column. It replaces FOIL for any size of factors and also runs backwards to factor trinomials.
How do I factor with the box method?+
Split the middle term using the ac method: find m and n with m·n = a·c and m + n = b. Put ax^2, mx, nx, and c in the box, then take the greatest common factor of each row and each column. Those GCFs are the binomial factors.
Is the box method the same as FOIL?+
For two binomials the four cells are exactly the First, Outer, Inner, and Last products. The box method keeps working for trinomials and bigger polynomials, where FOIL does not apply.
Which cells are like terms?+
Cells on the same diagonal (from top-right to bottom-left) have the same degree when the terms are written in descending order, so they combine.
What if no ac pair exists?+
Then the trinomial does not factor over the integers. Its roots are irrational or complex; use the quadratic formula to find them.
Can the box handle fractions?+
Multiplying works with any coefficients. For factoring, the coefficients are scaled to whole numbers first and the scale factor is shown in front of the answer.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
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Last updated: September 4, 2026