Here's a worked out solution, by completing the square.

If 5n + 33 divides n^2 - 17, then it also divides

25*(n^2 - 17)

= 25n^2 - 425

= (25n^2 - 1089) + 664

= (5n + 33)*(5n - 33) + 664

Note that 25 is relatively prime to 5n + 33, so this is the case if and only if 5n + 33 divides 664.

664 = 2*2*2*83,

so 5n + 33 must equal 1 or 2 or 4 or 8 or 83 or 166 or 332 or 664,

or -1 or -2 or -4 or -8 or -83 or -166 or -332 or -664

Clearly, if 5n + 33 is positive in ends in 3 or 8.

if 5n + 33 is negative it ends in 2 or 7

so, the only solutions are 5n + 33 = 8 or 83 or -2 or -332

In other words, n = -5 or 10 or -7 or -73

*Edited on ***May 23, 2014, 11:03 pm**