Find the coefficient of $x^8$ in $(x^2 + x^3 + x^4 + x^5)^5$.
I pulled the $x^2$ out to make it $$\left[x^2(1 + x + x^2 + x^3)\right]^5$$ and then $$x^{10}(1 + x + x^2 + x^3)^5$$ But then when I subtract the $x^{10}$ from the $x^8$ I get $x^{-2}$. Can I still find this coefficient or does it not work? Thanks
What you’ve discovered is that there is no $x^8$ term in $(x^2+x^3+x^4+x^5)^5$: the smallest exponent that appears when you multiply everything out is $10$. So the coefficient of $x^8$ is ... ?