Let $S_{1} \equiv x^{2}+y^{2}+2 x+3 y+1=0$
and $S_{2} \equiv x^{2}+y^{2}+4 x+3 y+2=0$,
then equation of common chord is $S_{2}-S_{1}=0$
$\Rightarrow 2 x+1=0$
Here, $ C_{1}\left(-1,-\frac{3}{2}\right), r_{1}=\frac{3}{2}=C_{1} P$
and $C_{2}\left(-2,-\frac{3}{2}\right), r_{2}=\frac{\sqrt{17}}{2}$
$C_{1} M=$ Perpendicular distance from $C_{1}$ to the common chord $2 x+1=0$
$\Rightarrow C_{1} M=\frac{|-2+1|}{\sqrt{2^{2}}}=\frac{1}{2}$
Now, $P Q=2 P M =2 \sqrt{\left(C_{1} P\right)^{2}-\left(C_{1} M\right)^{2}} $
$= 2 \sqrt{\left(\frac{3}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}=2 \sqrt{\frac{9}{4}-\frac{1}{4}}$
$= 2 \sqrt{2}$
