Factorize the given algebraic expression: $4{x^2} - 6xy$

Hint:Factors are numbers or algebraic expressions that divide another number or algebraic expression completely without leaving any remainder. We have to find factors of the given algebraic expression that are to take like terms together and use a grouping method of factorization. Taking common terms such as numbers and variables will help us to get the answer.

Complete solution step by step:
Firstly we write down the algebraic expression given in the question:
$4{x^2} - 6xy$

As we can see we are given two variables in the form of $x\,{\text{and}}\,y$ so we have to apply the grouping method of finding factors of an expression where we take like terms together and common numbers until we reach to an irreducible form. The separate components of this irreducible form will be the factors of the expression.

Applying this method to our expression we can see that we have $x$ as a common variable and 2 as a common number both of which divides the number so grouping the two we take $2x$ as a common factor and we have

$4{x^2} - 6xy = 2x(2x - 3y)$--equation (1)
As we can see we have broken the expression into two parts making it irreducible further because nothing can be taken as a common term between the two so we can say that

$2x$ is a factor of the given expression and
$\left( {2x - 3y} \right)$ is also a factor of the given expression and the factorized form can be written as this equation (1).

Note: Both the factors here in this question divide the given algebraic expression completely and this phenomenon can be easily understood by taking simple numbers like $6 = 2 \times 3$ which is written in a factorized form and $2,3$ are factors of the number dividing it completely without leaving any remainder.