Answer
$(a)\quad$ Zeros: $\quad -2 \quad 1\pm i\sqrt{3} $
$(b)\quad P(x)=(x+2)(x-1-i\sqrt{3} )(x-1+i\sqrt{3} )$
Work Step by Step
Complete Factorization Theorem (p.287)
$P$ factors into $n$ linear factors : $P(x)=a(x-c_{1})(x-c_{2})\cdots(x-c_{n})$
where $a$ is the leading coefficient of $P$ and $c_{1}, c_{1}, \ldots, c_{n}$ are the zeros of $P$.
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$(a)$
Recognize a sum of cubes
$x^{3}+2^{3}=0$
$ \begin{array}{lll}
(x+2)(x^{2}-2x+4) & =0 & \\
x+2=0 & or & x^{2}-2x+4=0\\
& & \\
x=-2 & & x=\frac{2\pm\sqrt{4-4(1)(4)}}{2}\\
& & \\
& & x=\frac{2\pm i\sqrt{12}}{2}\\
& & \\
& & x=\frac{2\pm 2i\sqrt{3}}{2}\\
& & \\
& & \\
& & x=1\pm i\sqrt{3} \\
& &
\end{array}$
Zeros: $-2 \quad 1\pm i\sqrt{3} $
$(b)$
The leading coefficient is 1,
$P(x)=(x+2)(x-1-i\sqrt{3} )(x-1+i\sqrt{3} )$
