#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)#
