First, let's expand the left side of the inequality:

(k+4)(k-4) - 6k + 5
= k^2 - 4k + 4k - 16 - 6k + 5
= k^2 - 6k - 11

Now let's expand the right side of the inequality:

(k-3)^2 + 4
= k^2 - 6k + 9 + 4
= k^2 - 6k + 13

So the inequality becomes:

k^2 - 6k - 11 > k^2 - 6k + 13

Subtracting k^2 and adding 6k on both sides, we get:

-11 > 13

This is not a true statement, so our initial assumption is incorrect. Therefore, there is no solution for this inequality.