It often helps to reword the question. Let's use this logic:
Def of prime number: a positive integer whose only integer factors are 1 and itself.
Note: for n=1, (n8+n+1)=3, which is a prime number.
Now, we must show that (n8+n+1) can be factored and that the factors cannot be 1 and (n8+n+1), except for n=1.
Yes, (n8+n+1) can be factored:
(n8 + n + 1) = (n2 + n + 1)*(n6 – n5 + n3 – n2 + 1)
Now, to determine that (n8+n+) is not prime, we must show that these factors are never both 1 and the number (n8+n+1) itself, except for n=1.
For positive integers n, the factor (n2+n+1) is never equal to 1 and it is never equal to (n8+n+1), except when n=1, since that would mean that n2=n8.
Therefore (n8+n+1) is not a prime number (except for n=1).