College Algebra, Chapter 1, 1.5, Section 1.5, Problem 66

Find all solutions, real and complex of the equation $\displaystyle x^3 + 3x^2 + 9x + 27 = 0$


$\begin{equation} \begin{aligned} x^3 + 3x^2 + 9x + 27 =& 0 && \text{Given} \\ \\ (x^3 + 3x^2) + (9x+ 27) =& 0 && \text{Group terms} \\ \\ x^2(x + 3) + 9 (x + 3) =& 0 && \text{Factor out } x^2 \text{ and } 9 \\ \\ (x^2 + 9)(x + 3) =& 0 && \text{Factor out } x^2 + 9 \\ \\ x^2 + 9 =& 0 \text{ and } x + 3 = 0 && \text{Zero Product Property} \\ \\ x =& \pm \sqrt{-9} \text{ and } x = -3 && \text{Solve for } x \\ \\ x =& \pm \sqrt{9i^2} \text{ and } x = -3 && \text{Recall that } i^2 = -1 \\ \\ x =& \pm 3i \text{ and } x = -3 && \text{Simplify} \end{aligned} \end{equation}$