Thank you very much.I think you may have copied the wrong equation for (9). The triangle inequality says that ##|a+b|\leq|a|+|b|##. Applying that to your equation we get:
[applying the triangle inequality three times in succession]
[applying (8) ]
$$=|z|^3+3|z|^2+2|z|+1<2^3+3\cdot 2^2+2\cdot 2+1=25$$
EDIT: Just saw Samy's post. I don't have the book but, based on that picture, it looks more likely a typo in that the ref to (9) should be to (10), rather than you miscopying it.