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Factorize: \[4{x^2} - 9{y^2} - 2x - 3y\].

Last updated date: 15th Jul 2024
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Hint: Here, we will use the concept of factorization. Factorization is the process in which a number is written in the form of its factors which on multiplication gives the original number. Here we will first rewrite the given expression in such a way that it takes the form of some algebraic identity. We will then expand the expression using a suitable algebraic identity. We will then factor out common terms to get the required factors of the expression.

Complete step by step solution:
The given expression is \[4{x^2} - 9{y^2} - 2x - 3y\].
We can write the above equation as
\[ \Rightarrow 4{x^2} - 9{y^2} - 2x - 3y = {\left( {2x} \right)^2} - {\left( {3y} \right)^2} - 2x - 3y\]
Now we will take \[ - 1\] common from the last two terms. Therefore, the above equation becomes
\[ \Rightarrow 4{x^2} - 9{y^2} - 2x - 3y = {\left( {2x} \right)^2} - {\left( {3y} \right)^2} - 1\left( {2x + 3y} \right)\]
Now we will use the algebraic identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]. Therefore, we get
\[ \Rightarrow 4{x^2} - 9{y^2} - 2x - 3y = \left( {2x - 3y} \right)\left( {2x + 3y} \right) - 1\left( {2x + 3y} \right)\]
We can see that the term \[\left( {2x + 3y} \right)\] is common in both the terms. Therefore taking common \[\left( {2x + 3y} \right)\] in the equation, we get
\[ \Rightarrow 4{x^2} - 9{y^2} - 2x - 3y = \left( {2x + 3y} \right)\left( {2x - 3y - 1} \right)\]

Hence, \[\left( {2x + 3y} \right),\left( {2x - 3y - 1} \right)\] are the factors of the given equation \[4{x^2} - 9{y^2} - 2x - 3y\].

Here we will note that we have to take the maximum common terms from the equation and we have to simplify the equation carefully to get the factors. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations. Generally in these types of questions, algebraic identities are used to solve and make the factors of the equation. Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation.