Answer
$x\in\{-\sqrt {6}i, \sqrt {6}i, -3i, 3i\}$ all with a multiplicity of $1$
Work Step by Step
$P(x)=x^4+15x^2+54$,
To solve for the polynomial,
Let's let $x^2=k$,
$P(x)=k^2+15k+54$,
factorize the trinomial $k^2+15k+54$,
(find factors of $54(1)=54$ whose sum is $15$):
($6$ and $9$)
$k^2+15k+54=k^2+6k+9k+54=k(k+6)+9(k+6)=(k+6)(k+9)$
Let's replace $x^2=k$ into the factorized trinomial,
$P(x)=(x^2+6)(x^2+9)$
thus, the zeros are:
$x^2+6=0, x=\pm\sqrt {6}i$ or $x^2+9=0, x=\pm3i$
$x\in\{-\sqrt {6}i, \sqrt {6}i, -3i, 3i\}$ all with a multiplicity of $1$
