#### Answer

2 rational solutions

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To evaluate the discriminant of the given equation, $
8x^2=-14x-3
,$ express first the equation in the form $ax^2+bx+c=0.$ Then identify first the value of $a,b,$ and $c.$ Then use the Discriminant Formula. If the value of the discriminant is less than zero, then there are $\text{
2 nonreal complex solutions
.}$ If the value is $0,$ then there is $\text{
1 distinct rational solution
.}$ If the value of the discriminant is a positive perfect square, then there are $\text{
2 rational solutions
.}$ Finally, if the value of the discriminant is positive but not a perfect square, there are $\text{
2 irrational solutions
.}$
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
8x^2=-14x-3
\\\\
8x^2+14x+3=0
.\end{array}
In the equation above, $a=
8
,$ $b=
14
,$ and $c=
3
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the value of the discriminant is
\begin{array}{l}\require{cancel}
(14)^2-4(8)(3)
\\\\=
196-96
\\\\=
100
.\end{array}
Since the discriminant is $\text{
a positive perfect square
,}$ then there are $\text{
2 rational solutions
.}$