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!

Well it's simple
we are multiplying every term by $x^3$ so coefficient of $x^{10}$ should be coeffiecent of $x^7$ in the expansion of $(x^2-3x^3-1)^6$
Now, divide three terms into two groups $((x^2-3x^3)+(-1))^6$ 
To get $x^7$, the only combination possible is when $x^2$ is raised to the power two and $-3x^3$ is rased to the power one and multiplied.So we get $6C3*(x^2-3x^3)^{(2+1)}*(-1)^{(6-(2+1))}$. I took 3 (2+1) because thats when i get the terms $x^7$.. for $x^7$ we get $6C3*3C2*(x^2)^2*(-3x^3)*(-1)^3$.Now multiply all the coefficent we get $20*3*-3*-1=180$
And you got your answer
