If $(2 x+y)^3-(x-2 y)^3=(x+3 y)\left[A x^2+B y^2+C x y\right]$, then what is the value of $(A+2 B+C)$ ?

10

We know that,

x³ - y³ = (x - y)(x² + y² + xy)

(2x + y)³ - (x - 2y)³ = (2x + y - x + 2y)[(2x + y)² + (x - 2y)² + (2x + y)(x - 2y)]

= (x + 3y)[4x² + y² + 4xy + x² + 4y² - 4xy + 2x² - 4xy + xy - 2y²]

= (x + 3y)[7x² + 3y² - 3xy]

So,

A = 7, B = 3 and C = - 3

So the value of(A + 2B + C) = 7 + 6 - 3 = 10