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Factorize the expression $(x+2y)^2-4(2x-y)^2$.
Given:
The given expression is $(x+2y)^2-4(2x-y)^2$.
To do:
We have to factorize the expression $(x+2y)^2-4(2x-y)^2$.
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.
$(x+2y)^2-4(2x-y)^2$ can be written as,
$(x+2y)^2-4(2x-y)^2=(x+2y)^2-[2(2x-y)]^2$ [Since $4=2^2$]
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,
$(x+2y)^2-4(2x-y)^2=(x+2y)^2-[2(2x-y)]^2$
$(x+2y)^2-4(2x-y)^2=[(x+2y)+2(2x-y)][(x+2y)-2(2x-y)]$
$(x+2y)^2-4(2x-y)^2=[(x+2y)+2(2x)-2(y)][(x+2y)-2(2x)+2(y)]$
$(x+2y)^2-4(2x-y)^2=(x+2y+4x-2y)(x+2y-4x+2y)$
$(x+2y)^2-4(2x-y)^2=(5x)(4y-3x)$
Hence, the given expression can be factorized as $5x(4y-3x)$.
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