We figure out the coefficient of x^2y^3 by figuring out the coefficients in both (x+y)^4 and (x+2y)^4.
Using the binomial theorem, the coefficient of x^2y^3 on the first expansion is
(4 choose 3)x^2y^3=6x^2y^3,
and the coefficient of x^2y^3 on the second expansion is
(4 choose 3)x^2(2y)^3=6x^2(4y^2)=24x^2y^3.
Thus, the coefficient in the sum of these two expansions is 6+24=30.
