P(x) is defined by the expression
[tex]P(x)=x^3+3x^2+6x[/tex]
Note
[tex]x^3+3x^2+6x=x(x^2+3x+6)\text{ }[/tex]
Therefore, one solution is 0.
The other two solutions come from
[tex]x^2+3x+6=0[/tex]
Apply the general solution in order to find complex solutions
[tex]\frac{-3\pm\sqrt{3^2-4(1)(6)}}{2(1)}=\frac{-3\pm\sqrt{-15}}{2}[/tex]
The solutions are
[tex]0,\frac{-3+i\sqrt{15}\text{ }}{2},\frac{-3-i\sqrt{15}}{2}[/tex]
We calculate the factor from the solutions, like this
[tex]x=\frac{-3+i\sqrt{15}}{2}\Rightarrow x+\frac{3}{2}-\frac{i\sqrt{15}}{2}=0[/tex][tex]x=\frac{-3-i\sqrt{15}}{2}\Rightarrow x+\frac{3}{2}+\frac{i\sqrt{15}}{2}=0[/tex]
The factor is
[tex]P(x)=x(x+\frac{3}{2}-\frac{i\sqrt{15}}{2})(x+\frac{3}{2}+\frac{i\sqrt{15}}{2})[/tex]
