Question

How do you factor `4+3xy-6y-2x` by grouping?

Answer 1

The answer would be (3y-2)(x-2) .

Explanation:

1. Arrange the equation with like terms near each other.
   3xy - 6y -2x + 4

2. Look for the GCF between the first two terms (i.e. a common factor, like y, that you can separate out). In this example we can see that 3xy and -6y have a GCF of y and 3. This is justified because y is a factor in both, and 6 is evenly divided by three, thus three is a factor of both. Take out the GCF.
   3y(x - 2) - 2x + 4

3. In order to group you need to be able to take out a GCF of two groups in order to make them the same base. For example, since I now have (x - 2) as a factor, in order to group I need to make sure that the -2x + 4 becomes (x - 2). I can right away see that -2x + 4 has a GCF of 2. So I take out the two to get: 2(-x + 2). However, note that (x-2) is different than (-x+2). Thus our work is not done.

4. Take out a negative GCF. This is a common solution to reverse signs in a factor when grouping. I simply add a negative sign to the 2 in 2(-x + 2), and because I am dividing by a negative, all the signs in the factor flip. Thus: -2(x - 2).

5. Now I have two factors that are the same, so I combine them to one, and then group the GCFs into a factor of their own. Thus:
   (3y - 2)(x - 2)