Answer
$\left\{ -1, 1, -i\sqrt{3}, i\sqrt{3} \right\}$
Work Step by Step
The 2 numbers whose product is $ac=
1(-3)=-3
$ and whose sum is $b=
2
$ are $\{
3,-1
.\}$ Using these two numbers to decompose the middle term of the given expression, $
x^4+2x^2-3=0
,$ then the factored form is
\begin{array}{l}\require{cancel}
x^4+3x^2-1x^2-3=0
\\\\
(x^4+3x^2)-(x^2+3)=0
\\\\
x^2(x^2+3)-(x^2+3)=0
\\\\
(x^2+3)(x^2-1)=0
.\end{array}
Equating each factor to zero, then,
\begin{array}{l}\require{cancel}
x^2+3=0
\\\\
x^2=-3
\\\\
x=\pm\sqrt{-3}
\\\\
x=\pm i\sqrt{3}
,\\\\\text{OR}\\\\
x^2-1=0
\\\\
x^2=1
\\\\
x=\pm\sqrt{1}
\\\\
x=\pm1
.\end{array}
Hence, the solutions are $
\left\{ -1, 1, -i\sqrt{3}, i\sqrt{3} \right\}
$.
