Start with some manipulation

= N^4+22N^2+121 - 49N^2

= (N^2+11)^2 - (7N)^2

= (N^2-7N+11) * (N^2+7N+11)

For this expression to evaluate to be noncomposite it is necessary that one of the terms must equal +/-1.

N^2-7N+11=1 has roots 2 and 5

N^2-7N+11=-1 has roots 3 and 4

N^2+7N+11=1 has roots -2 and -5

N^2+7N+11=-1 has roots -3 and -4

All the negative roots are to be discarded, since we are seeking positive values for N.  Testing each positive root:

2^4-27*2^2+121 = 29

3^4-27*3^2+121 = -41

4^4-27*4^2+121 = -55

5^4-27*5^2+121 = 71

-41 and -55 are to be discarded, since primes are defined to be positive, leaving 29 and 71 which are both prime.  Those two primes occur when N=2 or N=5.