Active band pass filter cutoff frequencies
This is all done by rearranging the formula the proper way or in a low-entropy form. I have derived the transfer function of this circuit in a previous post and the raw formula is the one you gave. However, your formula does not reveal the presence of a band-pass gain and this is often what is needed for designing such a filter. In the post I mentioned, I reorganized the transfer function with a \$Q\$ factor. You can rework your original expression a bit differently. With your transfer function, simply factor \$sC_1R_2\$ in the numerator and \$sR_1C_1\$ in the denominator. The \$s\$ goes away with \$C_1\$ and you now have a leading term with the dimension of a gain. This gain is not the mid-band gain you have when the magnitude plateaus:

\$H(s)=-\frac{sR_2C_1}{sR_1C_1}\frac{1}{(1+\frac{1}{sR_1C_1})(1+sR_2C_2)}=-\frac{R_2}{R_1}\frac{1}{(1+\frac{1}{sR_1C_1})(1+sR_2C_2)}\$

You know a pole is the inverse of the natural time constant in a first-order circuit. The above expression can thus be advantageously rewritten in its final low-entropy form where the gain appears together with the two poles:

\$H(s)=-H_1\frac{1}{(1+\frac{\omega_{p1}}{s})(1+\frac{s}{\omega_{p2}})}\$ in which \$H_1=\frac{R_2}{R_1}\$, \$\omega_{p1}=\frac{1}{R_1C_1}\$ and \$\omega_{p2}=\frac{1}{R_2C_2}\$.

If you want to unveil the mid-band gain where the magnitude plateaus, you have to resort to the formula I gave in the previous post. In this expression, the mid-band gain \$H_{bp}\$ was expressed as \$H_{bp}=\frac{R_2C_1}{R_1C_1+R_2C_2}\$.

The below Mathcad sheet shows all the transformations and compares the raw expression response versus the final one. Please note the presence of the inverted pole in the denominator. \$H_6(s)\$ includes the quality factor and predicts the mid-band gain nicely.



Expressing the transfer function this way lets you distinguish the gain and the two poles affecting this circuit. Your original formula was obviously correct, but you could not easily calculate the elements to match a design goal. You can learn more about the manipulations of transfer functions through various books like this one and this one.