finding open circuit voltage gain
Part of the current of the transconductance amplifier will flow back to the input rather than through \$R_2\$. So the KCL law in the output is not

$$g_mv_i + \frac{v_o}{R_2} = 0$$

As this will lead to your incorrect expression. But the correct KCL law would be:

$$g_mv_i + \frac{v_o}{R_2} + \frac{v_o-v_i}{R_f} = 0$$

Solving this will result in the exact expression given.