1. **State the problem:** Simplify the expression $(-3 + 3i\sqrt{3})x^5$. 2. **Identify the components:** The expression consists of a complex coefficient $-3 + 3i\sqrt{3}$ multiplied by $x^5$. 3. **Factor the coefficient:** Factor out the common factor 3: $$-3 + 3i\sqrt{3} = 3(-1 + i\sqrt{3})$$ 4. **Recognize the complex number:** The number $-1 + i\sqrt{3}$ can be expressed in polar form. Its magnitude is: $$|z| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$$ 5. **Find the argument (angle):** $$\theta = \arctan\left(\frac{\sqrt{3}}{-1}\right) = \arctan(-\sqrt{3})$$ Since the real part is negative and imaginary part positive, the complex number lies in the second quadrant, so: $$\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$$ 6. **Express in polar form:** $$-1 + i\sqrt{3} = 2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)$$ 7. **Rewrite the original expression:** $$(-3 + 3i\sqrt{3})x^5 = 3 \times 2 \left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) x^5 = 6 \left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) x^5$$ **Final simplified form:** $$6 \left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) x^5$$ This form is useful for further operations like powers or roots of complex numbers.