That you want to see a simple set of factors is not relevant. You essentially need to obtain the roots of this cubic polynomial in s, where K is a parameter, which would give you that factorization.
sroots = solve(s^3 + 10*s^2 + (21+K)*s + 4*K,s,'maxdegree',3)
sroots =
As you can see, they are fairly messy, and depending on the value of K, they may well be complex even if K is real. This is expected for any cubic polynomial, that we may find exactly two conjugate complex roots.
The factors now are just
prod(s-sroots)
ans =
We can expand that to show the result recovers your original polynomial.
simplify(expand(ans))
ans = 
Finally, why did not factor understand what you wanted to see? You gave it two variables s and K. How should factor possibly know that you wanted it to do what I just did, essentially, that s is the variable of interest, and K a symbolic parameter in the problem? Software cannot read your mind, at least, not yet.
