Answer
$(a)\quad$ Zeros: $\pm i\sqrt{3}$
$(b)\quad P(x)=(x-i\sqrt{3})^{2}(x+i\sqrt{3})^{2}$
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 perfect square
$(x^{2}+3)^{2}=0$
$ \begin{array}{ll}
x^{2}+3 & =0\\
x^{2}=-3 & \\
x=\pm i\sqrt{3} &
\end{array}$
Zeros: $\pm i\sqrt{3} \quad$ (both double multiplicity)
$(b)$
The leading coefficient is 1,
$P(x)=(x-i\sqrt{3})^{2}(x+i\sqrt{3})^{2}$
