$\Large f\left( x \right) ={ 8x }^{ 3 }-{ 12x }^{ 2 }+bx+d$

For variables $b$ and $d$ independent of $x$, we are given the function $f(x)$ as described above to have roots $-1\le\alpha ,\beta ,\gamma \le 1$ and that

$\alpha +\beta +\gamma =1+4\sin { \frac { \cos ^{ -1 }{ (\alpha ) } }{ 2 } } \times \sin { \frac { \cos ^{ -1 }{ (\beta) } }{ 2 } } \times \sin { \frac { \cos ^{ -1 }{ (\gamma ) } }{ 2 } }$

Evaluate $b+d$.

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