Simplify the given expression.

​$$(x-y)^3 + (x+y)^3 + 3(x-y)(x^2 -y^2) +3(x+y)(x^2-y^2)$$​​

Correct option is D

Given:

​$$(x-y)^3 + (x+y)^3 + 3(x-y)(x^2 -y^2) +3(x+y)(x^2-y^2)$$​​

Formula Used:

​$$(a+b)^3 =a^3 + b^3 +3ab(a+b)$$​​

​$$(a+b)^3 =a^3 - b^3 -3ab(a-b)$$​​

Solution:

​$$(x-y)^3 + (x+y)^3 + 3(x-y)(x^2 -y^2) +3(x+y)(x^2-y^2)$$​​

= $$(x-y)^3 + (x+y)^3 + (x^2 -y^2)(3(x-y) +3(x+y))$$​​

= $$(x-y)^3 + (x+y)^3 + (x^2 -y^2)(3x-3y +3x+3y)$$​​

= $$(x^3 + y^3 +3xy(x+y)) + (x^3 - y^3 - 3xy(x-y)) + (x^2 -y^2)(6x)$$​​

= $$2x^3 +3x^2y +3xy^2 + x^3 - 3x^2y +3xy^2 + 6x^3 -6xy^2$$​​

= $$2x^3 +6xy^2 + x^3 + 6x^3 -6xy^2$$​​

= $$8x^3$$​