#### Answer

$\text{Discriminant: }
0
\\\text{Number of solutions: }
1
\\\text{Type of solutions:
rational solution
}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To evaluate the discriminant of the given equation, $
x(9x+6)=-1
,$ identify first the values of $a,b,$ and $c$ in the quadratic expression $ax^2+bx+c.$ 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 $\text{
1 rational solution
.}$ 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 Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x(9x)+x(6)=-1
\\\\=
9x^2+6x=-1
.\end{array}
In the form $ax^2+bx+c=0,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
9x^2+6x+1=0
.\end{array}
In the equation above, $a=
9
,$ $b=
6
,$ and $c=
1
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the value of the discriminant is
\begin{array}{l}\require{cancel}
(6)^2-4(9)(1)
\\\\=
36-36
.\end{array}
Since the discriminant is $\text{
is equal to zero
,}$ then there is $\text{
1 rational solutions
.}$
Hence, the given equation has the following properties:
\begin{array}{l}\require{cancel}
\text{Discriminant: }
0
\\\text{Number of solutions: }
1
\\\text{Type of solutions:
rational solution
}
.\end{array}