Check for me plz!

$\phantom{=\qquad}(\ 3\sqrt6\ \text{cis}(\frac{\pi}{8})\ )(\ 2\sqrt5\ \text{cis}(\frac{7\pi}{6})\ )\\~\\ {=\qquad}6\sqrt{30}(\ \text{cis}(\frac{\pi}{8})\ )(\ \text{cis}(\frac{7\pi}{6})\ )\\~\\ {=\qquad}6\sqrt{30}(\ \cos(\frac{\pi}{8})+i\sin(\frac{\pi}{8})\ )(\ \cos(\frac{7\pi}{6})+i\sin(\frac{7\pi}{6})\ )\\~\\ {=\qquad}6\sqrt{30}(\ \cos(\frac{\pi}{8})\cos(\frac{7\pi}{6})\ +\ i\cos(\frac{\pi}{8})\sin(\frac{7\pi}{6})\ +\ i\sin(\frac{\pi}{8})\cos(\frac{7\pi}{6})\ -\ \sin(\frac{\pi}{8})\sin(\frac{7\pi}{6}) \ )\\~\\ {=\qquad}6\sqrt{30}(\ \cos(\frac{\pi}{8})\cos(\frac{7\pi}{6})\ -\ \sin(\frac{\pi}{8})\sin(\frac{7\pi}{6}) \ +\ i\cos(\frac{\pi}{8})\sin(\frac{7\pi}{6})\ +\ i\sin(\frac{\pi}{8})\cos(\frac{7\pi}{6}) \ )\\~\\ {=\qquad}6\sqrt{30}(\ { \color{blue} \cos(\frac{\pi}{8})\cos(\frac{7\pi}{6})\ -\ \sin(\frac{\pi}{8})\sin(\frac{7\pi}{6}) } \ +\ i {\color{purple}(\ \cos(\frac{\pi}{8})\sin(\frac{7\pi}{6})\ +\ \sin(\frac{\pi}{8})\cos(\frac{7\pi}{6}) \ )} \ )\\~\\ {=\qquad}6\sqrt{30}(\ { \color{blue} \cos(\frac{\pi}{8}+\frac{7\pi}{6}) } \ +\ i\ {\color{purple} \sin(\frac{\pi}{8}+\frac{7\pi}{6}) } \ )\\~\\ {=\qquad}6\sqrt{30}(\ { \cos(\frac{31\pi}{24}) } \ +\ i\ { \sin(\frac{31\pi}{24}) } \ )\\~\\ {=\qquad}6\sqrt{30}\ \text{cis}(\ \frac{31\pi}{24}\ )$

Check

You can use this formula for multiplying:  r₁cis(A₁) · r₂cis(A₂)  =  (r₁·r₂)cis(A₁ + A₂).

In your problem, you are correct in multiplying the two constants, but you should have added the angles, not multiplied them.