Solution:

Given expression is N =(18n² + 9n + 8)/n

To find out the integral solutions of N, we need to find the values of n which make the expression an integer.

Let us try to simplify the expression to identify the factors of N.

N = (18n² + 9n + 8)/n

= 18n + 9 + 8/n

= 18n + (8/n) + 9

Now, we can see that N will be an integer if and only if 8/n is an integer.

This is because 18n and 9 are always integers and their sum will also be an integer.

Therefore, we need to find the factors of 8 which are integers.

Factors of 8 are ±1, ±2, ±4, and ±8.

So, the possible values of n are ±1, ±2, ±4, and ±8.

Let us substitute each of these values of n in the expression for N and check whether it is an integer or not.

When n = 1, N = (18 + 9 + 8)/1 = 35, which is an integer.

When n = -1, N = (-18 - 9 + 8)/(-1) = 35, which is an integer.

When n = 2, N = (72 + 18 + 8)/2 = 49, which is an integer.

When n = -2, N = (-72 - 18 + 8)/(-2) = 49, which is an integer.

When n = 4, N = (288 + 36 + 8)/4 = 83, which is an integer.

When n = -4, N = (-288 - 36 + 8)/(-4) = 83, which is an integer.

When n = 8, N = (1296 + 72 + 8)/8 = 170, which is an integer.

When n = -8, N = (-1296 - 72 + 8)/(-8) = 170, which is an integer.

Therefore, there are 8 integral solutions of N possible, which are 35, 35, 49, 49, 83, 83, 170, and 170.