Answer:
Explanation:
Your equation has two complex roots, a=-3/4+sqrt(23)/4*i and b=-3/4-sqrt(23)/4*i. Calculate (x-a³/b³)*(x-b³/a³) and simplify it to get that expression. It will be a bit tedious, but the result is quite nice.
a³/b³=1001/1024+45*sqrt(23)/1024*i
b³/a³=1001/1024-45*sqrt(23)/1024*i
Notice that these are complex conjugates (which isn’t surprising, because a and b were complex conjugates). That’s why the result (x-a³/b³)*(x-b³/a³) will be a polynomial with real coefficients!
If you have complex conjugate roots c+di and c-di, you always get (x-(c+di))*(x-(c-di))=x²-2cx+c²+d², where c and d are real numbers.
It is -2c=-2*1001/1024=-1001/512, and
c²+d²=(1001/1024)²+(45*sqrt(23)/1024)²=1048576/1048576=1.