Question 1121240
At the end, I got what you got.
If one raises 2+i to the 4th power, the answer is -7+24 i
to the third power is 2+ 11i
squared is 3+4i
The constants, after using the exponents, are -7-12+27+12-20
The i s will be 24-66+36+6.  You did it right
{{{graph(300,300,-5,5,-50,50,x^4-6x^3+9x^2+6x-20)}}}  Your answer

another way, longer, uses the quadratic formula to find where 2+i and 2-i work
a=1
b=-4, because will divide by 2
sqrt(b^2-4ac)=2i, and when that is divided by 2, will give 1i as a result.  sqrt(16-20) does that, and since 4ac=20, c must equal 5
One polynomial is x^2-4x+5

The other polynomial has 
a=1
b=-2, so when divided by 2 will get -1
sqrt(b^2-4ac)  needs to be 2sqrt(5), because we are dividing the numerator by 2
b^2-4ac=20
4-4c=20
-4c=16
c=-4
x^2-2x-4 is the other polynomial

product is x^4-2x^3-4x^2
==========-4x^3+8x^2+16x
===============5x^2-10x-20
or x^4-6x^3+9x^2+6x-20