Electrical – Maximum voltage gain

Your result has no problem, and your procedure is all right. To verify this, you can substitute the \$Z_{b}\$

$$ Z_{b} = \beta r_{e}+[\frac{(\beta+1)+R_{c}/r_{o}}{1+(R_{C}+R_{E})/r_{o}}]R_{E} $$

into your result and into the textbook's result, and do some simplification, you'll find they give the same result. Cheers!

I think you just have an algebra mistake at the end. Starting from your last display equation: you can simplify the right-hand side to \$ (V_i - V_o) \frac{100}{120k} = \frac{V_i-V_o}{1.2k} \$. This is much bigger than the \$ \frac{V_i - V_o}{120k} \$ term on the left-hand side.

So you're left with \$ \frac{V_o}{0.5 k} = \frac{V_i - V_o}{1.2 k} \$, or more simply \$ 1.2 V_o = 0.5 (V_i - V_o) \$, giving \$V_o = 0.3 V_i \$. This agrees with the other analysis.