If -3+5i is a solution, then by the conjugate root theorem, -3-5i is also a solution. We find the polynomial by multiplying together the factors. If x = -4, then x + 4 is the factor. If x = -3+5i, then (x-(-3+5i)) is a factor, and so is (x-(-3-5i)). Simplifying those down gives us as the first factor as (x+3-5i) and the second as (x+3+5i). We can FOIL those 2 together to get their product, and then FOIL in x+4. FOILing the 2 complex factors together gives us [tex] x^2+3x+15ix+3x+9+15i-15ix-15i-25i^2 [/tex]. If we combine like terms and cross out things that cancel it's much easier than what it looks like there! It simplifies down to [tex] x^2+6x-25i^2 [/tex]. Since i^2 = -1, it simplifies further to [tex] x^2+6x-25(-1) [/tex] and finally, to [tex] x^2+6x+25 [/tex]. Now we will FOIL in x+4. [tex] (x^2+6x+25)(x+4)=x^3+6x^2+25x+4x^2+24x+100 [/tex]. Our final simplified third degree polynomial is [tex] x^3+10x^2+49x+100 [/tex]
