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 $5x(4y-3x)$.