Step 1:
$y^2+8x-6y+1=0$
$y^2-6y+1=-8x$
$y^2-6y+1+8=-8x+8$
$(y-3)^2=-8(x-1)$
Shifting the origin to $(1,3)$ by translation of axes
$X=x-1$
$Y=y-3$
Step 2:
$XY$ axis :
$y^2=-8x$
One parabola opens out towards the left.
$4a=8$
$a=2$
Axis : $Y=0$
Vertex : $(0,0)$
Focus : $(-2,0)$
Equation of directrix : $X=2$
Equation of LR : $X=-2$
$LR=4a=8$
$\Rightarrow 4\times 2=8$
Step 3:
$xy$ axis :
Axis : $y-3=0$
Vertex : $(1,3)$
Focus : $(-1,3)$
Equation of directrix : $x-1=2\Rightarrow 3$
Equation of LR : $x-1=-2\Rightarrow -1$
$LR=8$