Answer
$\begin{align*}
\text{Discriminant: }&
0
\\\text{Number of Real Solutions: }&
1
\end{align*}$
Work Step by Step
Using the properties of equality, the given equation, $
12x(x+1)=-3
,$ is equivalent to
\begin{align*}
12x(x)+12x(1)&=-3
&\text{ (use Distributive Property)}
\\
12x^2+12x&=-3
\\
12x^2+12x+3=0
\end{align*}
In the equation above $a=
12
,$ $b=
12
,$ 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}
&
12^2-4(12)(3)
\\&=
144-144
\\&=
0
\end{array}
Since the discriminant is equal to $0,$ then there is $1$ real solution. Hence,
\begin{align*}
\text{Discriminant: }&
0
\\\text{Number of Real Solutions: }&
1
\end{align*}