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Yes, you are on the right track
After multiplying by z^2 and rearranging you would have
z^4 + 5z^2 +4 = 0
this factors to
(z^2+1)(z^2+4) = 0
then z^ = -1 or z^2 = -4
there is not real number solution but if you use complex numbers where √-1 = i
z = ±i or z = ±2i