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

$$(2x+y)^3-(x-2y)^3=(x+3y)[Ax^2+By^2+Cxy]$$

$$\left[2x+y-\left(x-2y\right)\right]\left[\left(2x+y\right)^2+\left(2x+y\right)\left(x-2y\right)+\left(x-2y\right)^2\right]=(x+3y)[Ax^2+By^2+Cxy]$$

$$\left[x+3y\right]\left[4x^2+y^2+4xy+2x^2-3xy-2y^2+x^2+4y^2-4xy\right]=(x+3y)[Ax^2+By^2+Cxy]$$

$$\left(x+3y\right)\left[7x^2+3y^2-3xy\right]=(x+3y)[Ax^2+By^2+Cxy]$$

Comparing both sides,

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

$$A+2B+C\ =\ 7+2\left(3\right)-3$$ = 10

Hence, the correct answer is Option D