How to solve a quartic function with two unknown factors?

Question

Hi, been stuck on this one for a while. Any help would be appreciated.

 

z⁴ - 2z³ + az² + bz + 10 = 0, where a and b are real numbers. One of the solutions is 3+i.

 

I've tried solving for a and substituting it back in to find b, however, I haven't had any luck.

Answer 1

You substitute 3+i back into the eq, both the real

and imaginary parts are equal to zero. This gives

2 eqs. in 2 unknowns. 

I got: 8a+3b=-2 and  6a+b=-44 which gives 

a=-13 and b=34