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Factorize the expression $(2x+1)^2-9x^4$.
Given:
The given expression is $(2x+1)^2-9x^4$.
To do:
We have to factorize the expression $(2x+1)^2-9x^4$.
Solution:
Factorizing algebraic expressions:
Factorizing an algebraic expression means 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.
$(2x+1)^2-9x^4$ can be written as,
$(2x+1)^2-9x^4=(2x+1)^2-(3x^2)^2$ [Since $9x^4=(3x^2)^2$]
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,
$(2x+1)^2-9x^4=(2x+1)^2-(3x^2)^2$
$(2x+1)^2-9x^4=[2x+1+3x^2][2x+1-3x^2]$
$(2x+1)^2-9x^4=(3x^2+2x+1)(-3x^2+2x+1)$
Hence, the given expression can be factorized as $(3x^2+2x+1)(-3x^2+2x+1)$.
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