I am having difficulty with induction in several areas, however one question at a time.

Ex. if 6 divides n(n^2 + 5) for all n >=1 prove for n+1.

What I have so far is (k+1)((k+1)^2 +5)

= k^3 + 5k + 3k^2 + 3k + 6

which I have worked out to

k(k^2 +5) + 3k^2 + 3k + 6

I know 6| k(k^2 +5) but how to I prove 6 divides the rest?

Am I able to use

3k^2 mod 6 + 3k mod 6 + 6 mod 6 = 3+3+0 mod 6 = 0 mod 6 therefore also divisible by 6?