If we were doing it the long way, we would need to consider all the factors of 6:

1 & 6;
-1 & -6;
2 & 3;
-2 & -3

and also many factors of -12:

1 & -12;
2 & -6;
3 & -4
and all the negatives of these.

We could spend a long time finding the correct combination of factors if we use the long method.

Factoring by Grouping Method

Using grouping method, first we find 2 things, the:

(a) Product of the outer 2 terms of the trinomial
(b) Inner number of the trinomial

Then, the only "guess and check" we need to do is to look for 2 numbers whose:

• Product is the result of (a) above
• Sum is the inner term (from (b) above)

So in our 6x² + x − 12 example, we are looking for 2 terms whose:

(a) Product is 6x² × -12 = -72x² (multiply outer numbers)

and whose

(b) Sum is x (inner number).

We try some terms and easily get 9x and -8x.

These are correct since:

• (9x)(-8x) = -72x
• 9x + (-8x) = x

Now write the original expression replacing x with (9x − 8x), as follows:

6x² + x − 12 = 6x² + (9x − 8x) − 12

We now re-group the right-hand side:

6x² + (9x − 8x) − 12 = (6x² + 9x) − (8x + 12)

Now factor each of the bracketed terms:

(6x² + 9x) − (8x + 12) = 3x(2x + 3) − 4(2x + 3)

On the right-hand side, we notice that each term in brackets is the same, so we can combine them as follows:

3x(2x + 3) − 4(2x + 3) = (3x − 4)(2x + 3)

[What just happened?

If we have 3xA − 4A, we can factor A out of each term, and write (3x − 4)A. This is how grouping method works. You always end up with brackets that have the same terms inside, and these can be factored out.]

So our answer is:

6x² + x − 12 = (3x − 4)(2x + 3)

Always check your answer by multiplying it out!