I think you mean the complex number
-9 - 9√3 i
This number has modulus
|-9 - 9√3 i| = √((-9)² + (-9√3)²) = √324 = 18
and argument θ such that
tan(θ) = (-9√3) / (-9) = √3
Since -9 - 9√3 i falls in the third quadrant of the complex plane, we expect θ to be between -π and -π/2 radians, so that
θ = arctan(√3) - π = π/3 - π = -2π/3
Then the polar form is
18 (cos(-2Ï€/3) + i sin(-2Ï€/3))
and -2Ï€/3 is the same angle as 2Ï€ - 2Ï€/3 = 4Ï€/3, so the correct choice is
18 (cos(4Ï€/3) + i sin(4Ï€/3))
