Show that $\mathrm{gcd}(x+4,x-4)$ divides $8$ for all integers $x$. I want to prove that $\mathrm{gcd}(x-4,x+4)$ divides $8$ for all $x\in \mathbb{Z}$
Since they are both polynomials of degree $1$, it suggests that the $\mathrm{gcd}$ is a constant.
Using Euclidean Algorithm, I get: $(x+4) = 1(x-4) + 8$, so $\mathrm{gcd}(x-4,x+4)=\mathrm{gcd}(x-4,8)$ thus it will always divide $8$.
Is this the correct approach / use of EA for polynomials?
Thanks.

Your approach is correct. Another way of proving it is the following:
$$
d\mid x-4\text{ and }d\mid x+4\implies d\mid(x+4)-(x-4)=8.
$$
