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 ________.

The polynomial has the form $$f(x)=x^{4}+ax^{3}+bx^{2}+c$$ with real coefficients.

First use the given value at $$x=1$$:
$$f(1)=1+a+b+c=-9 \;\;\Longrightarrow\;\; a+b+c=-10 \;.-(1)$$

The derivative of $$f(x)$$ is
$$f'(x)=4x^{3}+3ax^{2}+2bx.$$

We are told that $$i\sqrt{3}$$ is a root of $$f'(x)=0$$. Because the coefficients of $$f'(x)$$ are real, its complex conjugate $$-i\sqrt{3}$$ is also a root. Hence the cubic dividend of $$f'(x)$$ must contain the factor $$x(x^{2}+3)$$:

$$f'(x)=4x^{3}+3ax^{2}+2bx = 4\,x\,(x^{2}+3).$$

Expanding the right-hand side gives $$4x^{3}+12x.$$ Match corresponding coefficients:

• Coefficient of $$x^{3}$$: already $$4$$ on both sides.
• Coefficient of $$x^{2}$$: $$3a=0\;\Longrightarrow\;a=0.$$
• Coefficient of $$x$$: $$2b=12\;\Longrightarrow\;b=6.$$

Substitute $$a=0,\;b=6$$ into equation $$(1)$$ to determine $$c$$:

$$0+6+c=-10\;\Longrightarrow\;c=-16.$$

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

Solve $$f(x)=0$$. Put $$y=x^{2}$$ to obtain the quadratic

$$y^{2}+6y-16=0.$$

Discriminant $$\Delta=6^{2}+4\!\cdot\!16=100,$$ so

$$y=\frac{-6\pm10}{2}\; \Longrightarrow\; y=2 \;\text{or}\; y=-8.$$

Hence

$$x^{2}=2\;\;\Longrightarrow\;\;x=\pm\sqrt{2},$$ $$x^{2}=-8\;\Longrightarrow\;\;x=\pm\,2i\sqrt{2}.$$

The four roots are therefore $$\alpha_{1}= \sqrt{2},\;\alpha_{2}=-\sqrt{2},\;\alpha_{3}= 2i\sqrt{2},\;\alpha_{4}=-2i\sqrt{2}.$$

Compute the sum of the squared moduli:

$$|\sqrt{2}|^{2}=2,\qquad|-\sqrt{2}|^{2}=2,$$ $$|2i\sqrt{2}|^{2}=(2\sqrt{2})^{2}=8,\qquad|-2i\sqrt{2}|^{2}=8.$$

Adding them gives $$|\alpha_{1}|^{2}+|\alpha_{2}|^{2}+|\alpha_{3}|^{2}+|\alpha_{4}|^{2}=2+2+8+8=20.$$

Therefore the required value is 20.