$(x+y)^2(x^3+y^3)^6 = (x^2+2xy+y^2)(x^3+y^3)^6 = (x^2+2xy+y^2)(...+Ax^2y^4+Bx^4y^2+Cx^3y^3)$. Clearly both $A$ and $B$ are zero, and $C$ is the coefficient of $x^3y^3$ in $(x^3+y^3)^6$, which is 6!
But then the coefficent of $x^4y^4$ in $(x+y)^2(x^3+y^3)^6$ would be $2*6!$ which is way too big.
$0$ because a degree of $x^4y^4$ is $8$, but $\deg{(x+y)^2(x^3+y^3)^6}=20$ and $(x+y)^2(x^3+y^3)^6$ is homogeneous.