How do you factor by grouping \[3ax-3bx-ay+by\]?

Hint: In order to find the solution of the given question that is how to factor by grouping \[3ax-3bx-ay+by\], take common factors using the distributive property. Here factoring a polynomial involves writing it as a product of two or more polynomials. It reverses the process of polynomial multiplication. And factoring by grouping means factoring polynomials by taking common factors using the concept of GCD and Distributive property.

Complete step-by-step answer:
According to the question, given expression in the question is as follows:
\[3ax-3bx-ay+by\]
In order to factorise the above expression by grouping, we will follow the following steps:
Step-1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.
\[3ax-3bx-ay+by\]
We can clearly see that in this case we have two terms in common, that is \[x\]is common among the first two terms and \[y\]is common among the last two terms.
Step-2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together, we get:
\[\Rightarrow \left( 3ax-3bx \right)+\left( -ay+by \right)\]
Step-3: Factor out the GCF from each of the two groups. In the second group, you have a choice of factoring out a positive or negative number. To determine whether you should factor out a positive or negative number, you need to look at the signs before the second and fourth terms. If the two signs are the same (both positive or both negative) you need to factor out a positive number. If the two signs are different, you must factor out a negative number. In this problem, the signs in front of the \[3bx\] and \[by\] are different, so we need to factor out a negative \[y\] and for first bracket we need to factor out positive \[3x\], we get:
\[\Rightarrow 3x\left( a-b \right)-y\left( a-b \right)\]
Step-4: Notice that what is inside the parenthesis is a perfect match, so it is time for the 2 for 1 special. The one thing that the two groups have in common is \[\left( a-b \right)\], so you can factor out \[\left( a-b \right)\] leaving the following:
\[\Rightarrow \left( a-b \right)\left( 3x-y \right)\]
As we can see the remaining factors can be factored any further.
Therefore, after factoring by grouping the given expression \[3ax-3bx-ay+by\] we get the final answer as \[\left( a-b \right)\left( 3x-y \right)\].

Note: Students make mistakes while taking the factors in common, they usually ignore the sign of the terms. This ignoring the sign of the terms can lead to not getting matching terms inside the parentheses which further makes you rearrange the four terms and try again until you get a perfect match. So, to determine whether you should factor out a positive or negative number, you need to look at the signs before the second and fourth terms. If the two signs are the same (both positive or both negative) you need to factor out a positive number. If the two signs are different, you must factor out a negative number.