[tex]n^2+6n-27=\\
n^2-3n+9n-27=\\
n(n-3)+9(n-3)=\\
(n+9)(n-3)[/tex]
For the above product to be a prime number, one of the factors must be a prime number and the other must be equal to 1.
[tex]n+9=1\\
n=-8\\\\
-8-3=-11[/tex]
The first factor is equal 1 for [tex]n=-8[/tex], but the other is euqal -11, which is not a prime number.
[tex]n-3=1\\
n=4\\\\
4+9=13[/tex]
The second factor is equal 1 for [tex]n=4[/tex] and the first factor is equal 13, which is a prime number.
So, [tex]n^2+6n-27[/tex] is a prime number for [tex]n=4[/tex]
