What is wrong with the proof?

Algebra Level 3
  1. Let xx be a solution of x2+x+1=0x^{2} + x + 1 = 0
  2. Since xx is not 0, we can divide both sides by xx:
    • x+1+1x=0x + 1 + \frac{1}{x} = 0
  3. Substitute x+1=x2x + 1 = -x^{2}
    • 1x=x2\frac{1}{x} = x^{2}
  4. x=1x = 1
    • x=x3x = x^{3}
  5. 3=03 = 0
    • 12+1+1=01^{2} + 1 + 1 = 0
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