Help! Polynomials

First, let's expand out everything. I'm not going to show the steps, but we eventually get

$2z^4-32z^3+204z^2-608z+706=-8\\2z^4-32z^3+204z^2-608z+714=0$

Now, since every number is divisible by 2, we get factor out 2 to get

${2\left(z^{4}-16z^{3}+102z^{2}-304z+357\right)}=0$

Now, note that $z^{4}-16z^{3}+102z^{2}-304z+357=(z^{2}-8z+17)(z^{2}-8z+21)$

This means that we have the equation

$2\left(z^{2}-8z+17\right)\left(z^{2}-8z+21\right)=0$

Since the equation is equal to 0, we get two equations. We have

$z^{2}-8z+17=0,\qquad z^{2}-8z+21=0$

Using the quadratic equation on both formulas, we find 4 values of z. 

We have

$z=4+i\\ z=4-i\\ z=4+\sqrt{5}\cdot (i)\\ z=4+\sqrt{5}\cdot (-i)$

This was a very basic explenation of the problem. 

I can go in dephth more if you want to. 

Thanks! :)