Find multiplier of x^6 in (a + bx + cx^2)^4 (Multinomail theorem)

Question

Have a simple problem where I need to find multiplier of $x^6$ in $(2+x+3*x^2)^4$

I tried doing it like this:
$ a = 2; b = x; c = 3*x^2$
Using multinomial theorem :
Possible combinations that give me $x^6$ would be bbcc and accc
$\frac{4!}{0!2!2!} * 2^0*x^2*(3*x^2)^2 + \frac{4!}{1!0!3!} * 2^1*x^0*(3*x^2)^3 = 270$
But the given answer for the multiplier in textbook is 63.
Where did I make the mistake? Any help appreciated, thanks.

Answer 1

You are correct see this: $$(2+x+3x^2)^4=81 x^8+108 x^7+270 x^6+228 x^5+289 x^4+152 x^3+120 x^2+32 x+16$$