Step 1:
$16x^2-9y^2+96x+36y-36=0$
$16x^2+96x-9y^2+36y=36$
Completing squares,
$16(x^2+6x+9)-9(y^2-4y+4)=36+144-36$
$16(x+3)^2-9(y-2)^2=144$
Step 2:
The above equation is divided by $144$
$\large\frac{(x+3)^2}{9}-\frac{(y-2)^2}{16}$$=1$
$a^2=9,b^2=16$
$\Rightarrow a=3,b=4$
Step 3:
Shifting the origin to $(-3,-2)$ by translation of axes $X=x+3$,$Y=y-2$
$\Rightarrow x=X-3,y=Y+2$
The equation reduces to $\large\frac{X^2}{9}-\frac{Y^2}{16}$$=1$
Transverse axes : $X$-axes (i.e) $Y=0\Rightarrow y-2=0$
Conjugate axes : $Y$-axes (i.e) $X=0\Rightarrow x+3=0$
Length of transverse axis =$2a=2\times 3=6$
Length of conjugate axis =$2b=2\times 4=8$