2. Let complex numbers $z, w$ satisfy $|z|=3,(z+\bar{w})(\bar{z}-w)=7+4 \mathrm{i}$, where $\mathrm{i}$ is the imaginary unit, $\bar{z}, \bar{w}$ represent the conjugates of $z, w$ respectively, then the modulus of $(z+2 \bar{w})(\bar{z}-2 w)$ is $\qquad$
$$
\begin{array}{l}
\text { Given that }(z+\bar{w})(\bar{z}-w)=z \bar{z}-w \bar{w}-(z w-\overline{z w})=|z|^{2}-|w|^{2}-(z w-\overline{z w}) \\
=7+4 \mathrm{i} \Rightarrow\left\{\begin{array} { l } 
{ | z | ^ { 2 } - | w | ^ { 2 } = 7 , } \\
{ z w - \overline { z w } = - 4 \mathrm { i } }
\end{array} \Rightarrow \left\{\begin{array} { l } 
{ 9 - | w | ^ { 2 } = 7 , } \\
{ z w - \overline { z w } = - 4 \mathrm { i } }
\end{array} \Rightarrow \left\{\begin{array}{l}
|w|^{2}=2, \\
z w-\overline{z w}=-4 \mathrm{i},
\end{array}\right.\right.\right.
\end{array}
$$

Thus, $(z+2 \bar{w})(\bar{z}-2 w)=z \bar{z}-4 w \bar{w}-2(z w-\overline{z w})=|z|^{2}-4|w|^{2}-2(z w-\overline{z w})$ $=9-8+8 \mathrm{i}=1+8 \mathrm{i}$, so the modulus of $(z+2 \bar{w})(\bar{z}-2 w)$ is $\sqrt{65}$.
$$