#### Answer

$x=\left\{ -\dfrac{1}{4},-\dfrac{3}{2} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
8x^2+14x+3=0
,$ express the equation in factored form. Next step is to equate each factor to zero (Zero Product Property). Finally, solve each equation.
$\bf{\text{Solution Details:}}$
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
8(3)=24
$ and the value of $b$ is $
14
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
2,12
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
8x^2+2x+12x+3=0
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(8x^2+2x)+(12x+3)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
2x(4x+1)+3(4x+1)=0
.\end{array}
Factoring the $GCF=
(4x+1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(4x+1)(2x+3)=0
.\end{array}
Equating each factor to zero (Zero Product Property), the solutions to the equation above are
\begin{array}{l}\require{cancel}
4x+1=0
\\\\\text{OR}\\\\
2x+3=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
4x+1=0
\\\\
4x=-1
\\\\
x=-\dfrac{1}{4}
\\\\\text{OR}\\\\
2x+3=0
\\\\
2x=-3
\\\\
x=-\dfrac{3}{2}
.\end{array}
Hence, $
x=\left\{ -\dfrac{1}{4},-\dfrac{3}{2} \right\}
.$