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
\\
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(x^2 + 9)(x + 3) =& 0
&& \text{Factor out } x^2 + 9
\\
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x^2 + 9 =& 0 \text{ and } x + 3 = 0
&& \text{Zero Product Property}
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x =& \pm \sqrt{-9} \text{ and } x = -3
&& \text{Solve for } x
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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}
$