Given that $(2x + y)^3 - (x + 2y)^3 = ( x - y) [A(x^2 +y^2)+Bxy]$, the value of (2A - B) is :

1

We know that,

 (a + b)³ = a³ + b³ + 3ab(a + b)

 (2x + y)³ - (x + 2y)³ = (x - y) [A(x² + y²) + Bxy]

= (2x)³ + y³ + 3 × 2x × y(2x + y) - (x)³ - (2y)³ - 3 × x × 2y (x + 2y) = (x - y) [Ax² + Ay² + Bxy]

= 8x³ + y³ + 12x²y + 6xy² - x³ - 8y³ - 6x²y - 12xy² = [Ax³ + Axy²+ Bx²y - Ax²y - Ay³ - Bxy²]

= 7x³ - 7y³ + 6x²y - 6xy² = Ax³ - Ay³ + (B - A)x²y + (A - B)xy²

By comparing LHS and RHS

= A = 7, (B - A) = 6, (A - B) = - 6

= (B - 7) = 6 

= B = 13

Now put the values of A and B in the required equation,

 (2A - B) = (2 × 7 - 13)  

=1