Finding a particular coefficient in a polynomial I'm trying to get the coefficient of $x^6$ of this polynomial product:
$$x^2(1+x+x^2+x^3+x^4+x^5)(1+x+x^2)(1+x^2+x^4).$$
I know with infinite series, you can use the closed form solution of the series to calculate the coefficient, but I haven't been able to figure out a way to calculate the coefficient for a finite series.
I hope there is a better way than doing the algebra.  

What is being asked is to find the coefficient of $x^4$ of
$$(1+x+x^2+x^3+x^4+x^5)(1+x+x^2)(1+x^2+x^4),$$
which, when multiplied by $x^2$ leads to the coefficient of $x^6$ overall. 
This can be obtained by expansion:
\begin{align}
P &= (1+x+x^2+x^3+x^4+x^5)(1+x+x^2)(1+x^2+x^4) \\
&= (1+x+x^2+x^3+x^4+x^5)(1+x+2 x^2 + x^3 + 2 x^4 + x^5 + x^6) \\
&= 1 + 2 x + 4 x^2 + 5 x^3 + 7 x^4 + \mathcal{O}(x^5) 
\end{align}
Now, $x^2 \, P = x^2 + 2 x^3 + 4 x^4 + 5 x^5 + 7 x^6 + \mathcal{O}(x^7)$.
