Find G. C. F. of $(8n^3 + 8n, 2n+1)$ I'm stuck with this problem, I divided $8n^3 + 8n$ by $2n+1$ and obtained $5$, so now my G. C. F is $\gcd(2n+1, -5)$.
What's next? I can't divide $2n+1$ by $-5$.

Well, you figured out that $\gcd(8n^3 +8n, 2n+1) = \gcd(2n+1, -5)$.
And as $\gcd(\pm a, \pm b) = \gcd(a b)$ we know $\gcd(8n^3 + 8n, 2n+1) = \gcd(2n+1,5)$.
As $5$ is prime then $\gcd(2n+1, 5)$ is either $1$ or $5$.
It is $5$ if $5|2n+1$.  ANd it is $1$ if $5\not\mid 2n+1$.
And $5|2n+1 \iff$
$2n+1 \equiv 0 \pmod 5 \iff$
$2n \equiv -1 \pmod 5 \iff$
$n \equiv 2 \pmod 5$.
So the answer is:
$\gcd(8n^3+8n, 2n+1) =\begin{cases} 5 &\text{if } n\equiv 2 \pmod 5\\1&\text{if } n\not\equiv 2 \pmod 5\end{cases}$
