a) 27y^3

b) 64y^3

c) 125y^3

d) 512y^3

correct answer is: b) 64y^3

Formula use: \left(a-b\right)^3=a^3-3a^2b+3ab^2-b^3

Explanation

Here, given expression is :
\left(x+2y\right)^3-{3.\left(x+2y\right)}^2\ \left(x-2y\right)+3\left(x+2y\right)\left(x-2y\right)^2-\left(x-2y\right)^3
Let,
\left(x+2y\right)=a
\left(x-2y\right)=b
Now, Simplify the expression :
\left(x+2y\right)^3-{3.\left(x+2y\right)}^2\ \left(x-2y\right)+3\left(x+2y\right)\left(x-2y\right)^2-\left(x-2y\right)^3
\rightarrow a^3-3.a^2.b+3.a.b^2-b^3
\therefore\left(a-b\right)^3
Now putting the value of ‘a’ & ‘b’, we get :
\{\left(x+2y\right)-\left(x-2y\right)\}^3
\rightarrow\left(x+2y-x+2y\right)^3
\rightarrow\left(4y\right)^3
\therefore 64y^3
Ans: 64y^3\ .