[tex](1+2x+3x^3)^{10}[/tex]
The only possible configurations of terms that contribute to generating the [tex]x^4[/tex] term in the expansion are given by the multinomial theorem to be
[tex]\left(\dbinom{10}{6,4,0}(1)^6(2x)^4(3x^3)^0+\dbinom{10}{8,1,1}(1)^8(2x)^1(3x^3)^1\right)x^4[/tex]
where
[tex]\dbinom{10}{k_1,\ldots,k_m}=\dfrac{10!}{k_1!\cdots k_m!}[/tex]
Simplifying a bit gives a coefficient of
[tex]\dbinom{10}{6,4,0}(2)^4+\dbinom{10}{8,1,1}(2)(3)[/tex]
[tex]16\dbinom{10}{6,4,0}+6\dbinom{10}{8,1,1}[/tex]
[tex]16\dfrac{10!}{6!4!0!}+6\dfrac{10!}{8!1!1!}[/tex]
[tex]16(5\times3\times2\times7)+6(10\times9)=3900[/tex]
