Generating function for modeling the number of ways to select five integers from 1 to n where no two are consecutive So, I guess divide it into six sections representing the difference between the numbers, so the first is $0\rightarrow n$, the second through fifth are $1\rightarrow n$ and the sixth is $0\rightarrow n$ again, and that becomes a generating function of $((1+x+x^2+\cdots+x^n)^2)((x^2+\cdots+x^n)^4)$. Does this make sense, and what coefficient represents what value of $n$?

I don't understand why this question is asked in the language of generating functions.  The easy way to solve the problem is to take any subset $\{a,b,c,d,e\}$ of $\{1,\dots,n-4\}$, and choose the subset $\{a,b+1,c+2,d+3,e+4\}$.  In particular, there are $\binom{n-4}{5}$  ways to do this.  We can recover the generating function using basic generating function knowledge:
$$\sum_{n\geq 0} \binom{n-4}{5}x^n =\sum_{n\geq 0} \binom{n-9+5}{5}x^n=x^9\sum_{n\geq 0} \binom{n-9+5}{5}x^{n-9}$$
Changing indices and recognizing the form of the generating function, we have 
$$=x^9\sum_{N\geq -9} \binom{N+5}{5}x^N=\frac{x^9}{(1-x)^6}.$$
