First find the factor pairs of 16: (1,16), (2,8), (4,4) are the only ones. Also note that the coefficient of k is 1, meaning in the simplification that all coefficients of k will be 1. Also it's a cubic equation and if it is to factorise, we can expect a term like k+c where c is a number. We can also expect something like k^2+ak+b as another factor where a and b are constants.
(k+c)(k^2+ak+b)=k^3+ak^2+bk+ck^2+ack+bc=k^3-k^2-14k-16=0
Now we look at the coefficients and constant term. For the constant bc=-16 which takes us back to the factor pairs. It's got to be just one of them. But we do need to take note of the minus sign.
We don't need to think about the k^3 term because it will cancel out.
k^2 term: a+c=-1, so c=-1-a or -(1+a)
k term: b+ac=-14
What possibilities do we have for b and c? One must be positive and one negative so we have:
(-1,16), (1,-16), (-2,8), (2,-8), (-4,4), (4,-4).
We've already been given the answer so let's see what happens when we use (-8,2).
a+2=-1 so a=-3, and we already have c=2 and b=-8 so we have all the numbers.
b+ac=-8-6=-14 which is true. We have consistency.
(k+c)(k^2+ak+b)=(k+2)(k^2-3k-8).
Now, I shortened this because we should have tried all the factor pairs as if we didn't already know the answer. But we would have found only one that fitted, because the other answers would have been inconsistent. Nevertheless, the method shows how we compare coefficients.