#### Answer

$x=\left\{ -\dfrac{1}{2},\dfrac{1}{2},-i, i \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
4x^4+3x^2-1=0
,$ use factoring. Then equate each factor zero and solve the variable using the Square Root Principle.
$\bf{\text{Solution Details:}}$
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the factored form of the equation above is
\begin{array}{l}\require{cancel}
(4x^2-1)(x^2+1)=0
.\end{array}
Equating each factor to zero (Zero Product Property) results to
\begin{array}{l}\require{cancel}
4x^2-1=0
\\\\\text{OR}\\\\
x^2+1=0
.\end{array}
Isolating the squared variable results to
\begin{array}{l}\require{cancel}
4x^2-1=0
\\\\
4x^2=1
\\\\
x^2=\dfrac{1}{4}
\\\\\text{OR}\\\\
x^2+1=0
\\\\
x^2=-1
.\end{array}
Taking the square root of both sides (Square Root Principle) results to
\begin{array}{l}\require{cancel}
x^2=\dfrac{1}{4}
\\\\
x=\pm\sqrt{\dfrac{1}{4}}
\\\\
x=\pm\sqrt{\left( \dfrac{1}{2} \right)^2}
\\\\
x=\pm\dfrac{1}{2}
\\\\\text{OR}\\\\
x=\pm\sqrt{-1}
\\\\
x=\pm i
\text{ (Note that $i=\sqrt{-1}$) }
.\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{1}{2},\dfrac{1}{2},-i, i \right\}
.$