$S_{1}: x^{2}+y^{2}-2 x+4 y+4=0$
Centre, $C_{1}( 1 ,-2)$ and $r_{1}= 1$
and $S_{2}: x^{2}+y^{2}+4 x-2 y+1=0$
Centre $ C_{2}(-2, 1 )$ and $r_{2}=2$
Distance between centres, $d$ is
$d=\sqrt{(1+2)^{2}+(-2-1)^{2}} $
$d=\sqrt{18}=3 \sqrt{2}$
$\because d > r_{1}+r_{2}$
$\therefore S_{1}$ and $S_{2}$ are not intersecting each other.
The length of transversal common tangent is
$L=\sqrt{d^{2}-\left(r_{1}+r_{2}\right)^{2}}=\sqrt{(3 \sqrt{2})^{2}-9}=\sqrt{9}$
$ L=3$ units