Factorize the expression $(2a-b)^2-16c^2$.

Given:

The given algebraic expression is $(2a-b)^2-16c^2$.

To do:

We have to factorize the expression $(2a-b)^2-16c^2$.

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.

$(2a-b)^2-16c^2$ can be written as,

$(2a-b)^2-16c^2=(2a-b)^2-(4c)^2]$             [Since $16=4^2$]

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,

$(2a-b)^2-16c^2=(2a-b)^2-(4c)^2$

$(2a-b)^2-16c^2=(2a-b+4c)(2a-b-4c)$

Hence, the given expression can be factorized as $(2a-b+4c)(2a-b-4c)$.