Factoring a polynomial doesn't have to be a guessing game. The box method for factoring quadratic polynomials provides a visual method for performing the necessary calculations. The standard form for a quadratic polynomial is ax^2 + bx + c; a, b and c represent integer coefficients. If you have a polynomial that is not in standard form, convert it to standard form using algebraic operations before beginning the box method.
Draw a large "X." Multiply the last term of the polynomial by the coefficient of the first term and write the result at the top of the X. Write the coefficient of the second term at the bottom of the X.
Find all the sets of factors for the number at the top of the X. If the number is -4, for example, possible factors are 2 and -2, 1 and -4 or -1 and 4. Write one number in each set of factors at the left of the X and the other number at the right.
Add each set of factors together and choose the factors that add up to the number at the bottom of the X. For example, if the number is 3, choose the factors -1 and 4. Circle these numbers.
Draw a square. Add horizontal and vertical lines through the center of the square to divide it into four boxes.
Write the first term of the polynomial in the upper left box. Write the third term of the polynomial in the bottom right box. Write the circled numbers from your "X" in the remaining boxes and add an "x" after each one -- it does not matter which number goes in which box.
Find the greatest common factor for the numbers in each row and column of the box and write the GCF at the top of the column or to the left of the row. For example, if 2x^2 and -1x are in the top row, write the common factor "x" to the left; if 4x and -2 are in the bottom row, write the common factor 2 to the left. Follow the same procedure for each column. Only use a negative factor if both numbers if the row or column are negative.
Use the terms on the top and to the left of the box to write the factored form of the polynomial. For example, if 2x and -1 are at the top of the box and x and 2 are to the left, the factored polynomial is (2x - 1)(x + 2).