Find all possible '$b$' for which  $f\left(x\right)=\sin 2x-8(b+2)\cos x-(4b^2+16b+6)x$  is monotonic decreasing  for $\mathrm{\forall }x\in R$  and has no critical point.

$f^{\mathrm{\prime }}\left(x\right)=2\cos 2x+8(b+2)\sin x-(4b^2+16b+6)$

$=2(1-2{\sin }^{2}x)+8(b+2)\sin x-(4b^2+16b+6)$

for monotonic decreasing and no critical points

$f^{\mathrm{\prime }}\left(x\right)<0\mathrm{\forall }\mathrm{x}\in \mathrm{R}$

hence  ${\sin }^{2}x-2(b+2)\Rightarrow \sin x+b^2+4b+1>0$

$\Rightarrow b\in \left(\right(\sqrt{3}-1),\mathrm{\infty })$