We have:
z=-5+4i. Square z: z^2=25-40i-16=9-40i;
10z=-50+40i;
and z^2+10z=-41 so z^2+10z+41=0.
(z^2+10z)^2=1681=z^4+20z^3+100z^2.
z^4+9z^3+35z^2-z+4=
(z^4+20z^3+100z^2)-11z^3-65z^2-z+4=
1681-11z^3-65z^2-z+4.
z^3=(9-40i)(-5+4i)=-45+236i+160=115+236i;
-11z^3=-1265-2596i.
z^2=-41-10z;
-65z^2=2665+650z=2665+65(-50+40i)=-585+2600i.
-11z^3-65z^2-z+4=-1850+4i=-1850+z+5-z+4=-1841.
z^4+9z^3+35z^2-z+4=1681-1841=-160.