<s>[INST] Find coefficient for $x^{10}$ in $x^3(x^2-3x^3-1)^6$ I'm trying to solve the following problem:

Find the coefficient for  $x^{10}$ in $x^3(x^2-3x^3-1)^6$.

Can I use the multinomial theorem to solve it? I'm unsure how to start..
Thanks!
 [/INST] Definition: 
$[x^n]f(x)~$ means the coefficient of $~x^n~$ of the Taylor expansion of $~f(x)~$ at $~x=0~$ . 
Calculation:
$\displaystyle [x^{10}]x^3(x^2-3x^3-1)^6 = [x^{7}](x^2-3x^3-1)^6 = [x^{7}]\sum\limits_{k=0}^6 {\binom 6 k}x^{2k}(3x-1)^k = $
$\displaystyle =\sum\limits_{k=0}^6 {\binom 6 k}[x^{7-2k}](3x-1)^k =\sum\limits_{~~~~~k=0\\0\leq 7-2k\leq k}^6 {\binom 6 k}[x^{7-2k}](3x-1)^k$
$\displaystyle =\sum\limits_{k=3}^3 {\binom 6 k}[x^{7-2k}](3x-1)^k = {\binom 6 3}[x^{1}](3x-1)^3 =20\cdot 3{\binom 3 1}=180$
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