I've been doubting myself on this problem quite a bit.

The problem asks to prove that n^2 - 7n + 12 is nonnegative whenever n is an integer with n >= 3

When i went to prove this I made my basis step

p(3) = (3)^2 - 7(3) + 12 >= 0

p(3) = 9 - 21 + 21 >= 0

p(3) = 0 >= 0

after that, I assumed p(k)

p(k) = k^2 - 7k +12 >= 0

p(k) = (k-4)(k-3) > = 0

and

p(k+1) = (k+1)^2 - 7(k+1) + 12 >= 0

p(k+1) = k^2 - 5k + 6 >= 0

p(k+1) = (k-2)(k-3) >= 0

This is where I start to doubt myself.

(k-2)(k-3) >= (k-4)(k-3) >= 0

q.e.d.

(k-2)(k-3) >= 0

If I could get some feedback on this, it would be greatly appreciated.