You can do this by grouping:
[tex]3x^3+9x^2-2x-6=3x^2(x+3)-2(x+3)=(3x^2-2)(x+3)[/tex]
You could continue to factor [tex]3x^2-2=(\sqrt3\,x)^2-(\sqrt2)^2[/tex] as a difference of squares so that
[tex]3x^3+9x^2-2x-6=(\sqrt3\,x+\sqrt 2)(\sqrt3\,x-\sqrt2)(x+3)[/tex]
[tex]3x^3+9x^2-2x-6=3\left(x+\sqrt{\dfrac23}\right)\left(x-\sqrt{\dfrac23}\right)(x+3)[/tex]
