$$\begin{gathered} \mathrm{Equation} \mathrm{of} \mathrm{the} \mathrm{hyperbola} \mathrm{passing} \mathrm{through} \mathrm{the} \mathrm{point} (1,-1) \mathrm{and} \mathrm{having} \mathrm{asymptotes} \mathrm{x}+2\mathrm{y}+3=0 \mathrm{and} \\ 3\mathrm{x}+4\mathrm{y}+5=0 \mathrm{is} \end{gathered}$$

$$\begin{gathered} Given asymptotes of hyperbola are x+2y+3=0 and 3x+4y+5=0 \\ Now e\mathrm{quation} \mathrm{of} \mathrm{hyperbola} \mathrm{is} (\mathrm{x}+2\mathrm{y}+3)(3\mathrm{x}+4\mathrm{y}+5)=k .....\left(1\right) \\ \left(1\right) \mathrm{passing} \mathrm{through} (1,-1)\Rightarrow k=8 \\ from \left(1\right) equation of hyperbola is (\mathrm{x}+2\mathrm{y}+3)(3\mathrm{x}+4\mathrm{y}+5)=8 \end{gathered}$$