#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)#