Factorize the expression $4(xy+1)^2-9(x-1)^2$.

Given:

The given algebraic expression is $4(xy+1)^2-9(x-1)^2$.

To do:

We have to factorize the expression $4(xy+1)^2-9(x-1)^2$.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. 

An algebraic expression is factored completely when it is written as a product of prime factors.

$4(xy+1)^2-9(x-1)^2$ can be written as,

$4(xy+1)^2-9(x-1)^2=[2(xy+1)]^2-[3(x-1)]^2$             [Since $4=2^2, 9=3^2$]

Here, we can observe that the given expression is a difference of two squares. So, by using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize the given expression. 

Therefore,

$4(xy+1)^2-9(x-1)^2=[2(xy+1)]^2-[3(x-1)]^2$

$4(xy+1)^2-9(x-1)^2=[2(xy+1)+3(x-1)][2(xy+1)-3(x-1)]$

$4(xy+1)^2-9(x-1)^2=[2xy+2+3x-3][2xy+2-3x+3]$

$4(xy+1)^2-9(x-1)^2=(2xy+3x-1)(2xy-3x+5)$

Hence, the given expression can be factorized as $(2xy+3x-1)(2xy-3x+5)$.

Akhileshwar Nani

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Updated on: 09-Apr-2023

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