f(s) = 4s^5 + 8s^4 + 5s^3 + 10s^2
f(s) = s^2(4s^3 + 8s^2 + 5s + 10)
After factoring out s^2 we are left with a polynomial of degree 3 to factorise g(s) = 4s^3 + 8s^2 + 5s + 10. This can be done using the factor theorem.
After testing some integers it can be found that:
g(-2) = 0
Hence (s+2) is a factor of g(s) and can be factored out by long division. This gives the result:
g(s) =(s+2)(4s^2 + 5)
4s^2+5 can be factorised further using the quadratic formula.
s = (-0 +-sqrt(0^2 - 4 xx 4 xx 5))/(2 xx 4)
s = +-sqrt(-80)/8
s = +-isqrt(5)/2
Hence
g(s) = (s+2)(s + isqrt(5)/2)(s - isqrt(5)/2)
And to answer your question:
4s^5 + 8s^4 + 5s^3 + 10s^2 = s^2(s+2)(s + isqrt(5)/2)(s - isqrt(5)/2)