The 
method is an algorithm for factoring
quadratic polynomials of the form 
with integer coefficients.
As its name suggests, the crux of the algorithm is to consider the multiplicative
factors of the product of
the coefficients 
and 
.
More precisely, the goal is to find an integer pair 
and 
satisfying 
and 
simultaneously, whereby one can
rewrite 
in the form
 |
(1)
|
and to factor the remaining four-term polynomial by grouping into a product of linear factors with integer coefficients.
For example, consider the polynomial 
having coefficients 
, 
,
and 
. To begin the 
factorization, consider the product 
. By observation, 
while 
; in particular, this guarantees that 
can be rewritten so that
 |
(2)
|
This four-term expression for 
can be factored by grouping:
 |
(3)
|
and so
 |
(4)
|
One can easily see that the above method generalizes to certain polynomials of the form 
for positive integers
, though the result will be a factorization
into pairs of polynomials of degree 
which aren't necessarily linear.
This procedure is an alternative to the more straightforward utilization of the quadratic formula and has a number of drawbacks.
For example, finding 
and 
hinges on observation and/or guess-and-check;
this can be particularly problematic when the product 
has a large number of factors. Moreover, while the quadratic
formula illustrates immediately the existence of irrational
and/or imaginary roots,
the 
method often disguises such behavior
and thus requires a degree of "pre-processing," e.g., by analyzing the
polynomial discriminant.
