Let $f ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c$ be a polynomial with real coefficients such that $f ( 1 ) = - 9$. Suppose that $i \sqrt { 3 }$ is a root of the equation $4 x ^ { 3 } + 3 a x ^ { 2 } + 2 b x = 0$, where $i = \sqrt { - 1 }$. If $\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }$, and $\alpha _ { 4 }$ are all the roots of the equation $f ( x ) = 0$, then $\left| \alpha _ { 1 } \right| ^ { 2 } + \left| \alpha _ { 2 } \right| ^ { 2 } + \left| \alpha _ { 3 } \right| ^ { 2 } + \left| \alpha _ { 4 } \right| ^ { 2 }$ is equal to $\_\_\_\_$ .
Let $f ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c$ be a polynomial with real coefficients such that $f ( 1 ) = - 9$. Suppose that $i \sqrt { 3 }$ is a root of the equation $4 x ^ { 3 } + 3 a x ^ { 2 } + 2 b x = 0$, where $i = \sqrt { - 1 }$. If $\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }$, and $\alpha _ { 4 }$ are all the roots of the equation $f ( x ) = 0$, then $\left| \alpha _ { 1 } \right| ^ { 2 } + \left| \alpha _ { 2 } \right| ^ { 2 } + \left| \alpha _ { 3 } \right| ^ { 2 } + \left| \alpha _ { 4 } \right| ^ { 2 }$ is equal to $\_\_\_\_$ .