easy!

Since (2x+3y)^2 is equal to 4x^2+9y^2+12xy. It implies (4x^2+9y^2+12xy)^6 = (2x + 3y) ^ 12 since it will be expanded in 13 ways the answer is 13

Since $4x^{2}+9y^{2}+12xy$ is homogenous equation, $(4x^{2}+9y^{2}+12xy)^{6}$ . Now on expanding, its obvious we will get a homogenous equation of degree 12 which is to be distributed in x and y. So total ways=13

(Since none of coefficients is negative, we will get each of 13 distributions!)