Let $f(x) = x^{4} + ax^{3} + bx^{2} + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^{3} + 3ax^{2} + 2bx = 0$ 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 $|\alpha_{1}|^{2} + |\alpha_{2}|^{2} + |\alpha_{3}|^{2} + |\alpha_{4}|^{2}$ is equal to ______.

Let f(x) = x⁴ + ax³ + bx² + c be a polynomial with real coefficients such that f(1) = -9. Suppose that i√3 is a root of the equation 4x³ + 3ax² + 2bx = 0 where i = √-1. If α₁, α₂, α₃ and α₄ are all the roots of the equation f(x) = 0, then |α₁|² + |α₂|² + |α₃|² + |α₄|² is equal to ______.

$f(x) = x^{4} + ax^{3} + bx^{2} + c$

$a, b, c \in \mathbb{R},\quad f(1) = -9$

$i\sqrt{3} \text{ is a root of } 4x^{3} + 3ax^{2} + 2bx = 0$

$\text{To find: } |\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$

$\text{Argand plane}$

$\text{Plan: } 1.\ \text{use the root } i\sqrt{3} \Rightarrow a,\, b$

$2.\ f(1) = -9 \Rightarrow c$

$3.\ \text{solve } f(x) = 0 \text{ and add } |\alpha|^2$

$4x^{3} + 3ax^{2} + 2bx = x(4x^{2} + 3ax + 2b)$

$4(i\sqrt{3})^{2} + 3a(i\sqrt{3}) + 2b = 0$

$4(-3) + 3a\,i\sqrt{3} + 2b = 0$

$(-12 + 2b) + 3\sqrt{3}\,a\,i = 0$

$\text{Real: } -12 + 2b = 0 \Rightarrow b = 6$

$\text{Imag: } 3\sqrt{3}\,a = 0 \Rightarrow a = 0$

$f(1) = 1 + a + b + c = -9$

$1 + 0 + 6 + c = -9 \Rightarrow c = -16$

$f(x) = x^{4} + 6x^{2} - 16$

$\text{Let } y = x^{2}:\quad y^{2} + 6y - 16 = 0$

$y = \dfrac{-6 \pm \sqrt{36 + 64}}{2} = \dfrac{-6 \pm 10}{2}$

$y = 2 \ \text{or}\ y = -8$

$x = \pm\sqrt{2},\quad x = \pm 2\sqrt{2}\,i$

$|\sqrt{2}|^{2} = 2,\qquad |2\sqrt{2}\,i|^{2} = 8$

$2 + 2 + 8 + 8 = 20$

$|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2 = 20$