[tex]f = x^3+2x^2+ax+1 \\ \\ \text{Notam radacinile cu: }x_1=A,\,x_2=\overline{A},\, x_3 =\pm |A| \\ \\ x_1\cdot x_2\cdot x_3 = -1 \Rightarrow A\cdot \overline{A}\cdot (\pm |A|) = -1 \Rightarrow \\ \Rightarrow |A|^2\cdot (\pm |A|) = -1 \Rightarrow \\ \\ \Rightarrow \pm |A|^3 = -1 \Rightarrow x_3< 0\text{ pentru ca }|A|^3 = -1 \text{ nu are sens}[/tex]
[tex]\Rightarrow x_3 = -|A| \\ \\\Rightarrow -|A|^3 = -1 \Rightarrow |A|^3 = 1 \Rightarrow (|A|-1)(|A|^2+|A|+1) = 1 \Rightarrow \\ \\ \Rightarrow |A| = 1,\quad \text{Pentru ca }|A|^2+|A|+1\text{ nu are solutii reale}.[/tex]
[tex]x_1+x_2+x_3 = -2 \Rightarrow A+\overline{A}-|A| = -2 \Rightarrow \\ \Rightarrow A+|\overline A| -1 = -2 \Rightarrow A+|\overline{A}| = -1 \\ \\ x_1x_2+x_1x_3+x_2x_3 = -1 \Rightarrow \\ \Rightarrow A\cdot \overline{A}-A\cdot |A|-\overline{A}\cdot |A| = a \Rightarrow |A|^2 - |A|(A+\overline{A}) = a \Rightarrow \\ \\ \Rightarrow 1^2 - 1\cdot (A+\overline{A}) = a\Rightarrow 1-(-1) = a \Rightarrow \boxed{a = 2}[/tex]
