<s>[INST] If $z^2+z+2=0$, then $z^2 + \frac{4}{z^2} = -3$ 
If $z^2+z+2=0$, for $z\in\mathbb C$, demonstrate: 
  $$z^2 + \frac{4}{z^2} = -3$$

$z^2 = - 2 - z$, but it didn't help me.
Is there any other elegant solution?
 [/INST] $$z^2+\frac{4}{z^2}+3$$
$$=\frac{z^4+3z^2+4}{z^2}$$
$$=\frac{(z^2+z+2)(z^2-z+2)}{z^2}$$
$$=0$$
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