#### Answer

$c=9$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To find the value of $
c
$ such that the given equation, $
4m^2+12m+c=0
,$ will have $1$ rational solution, equate the discriminant to $0$ and solve for the variable.
$\bf{\text{Solution Details:}}$
In the equation above, $a=
4
,$ $b=
12
,$ and $c=
c
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the discriminant is
\begin{array}{l}\require{cancel}
(12)^2-4(4)(c)
\\\\=
144-16c
.\end{array}
Equating the discriminant to $0$ so that the given equation will have $1$ rational solution, then
\begin{array}{l}\require{cancel}
144-16c=0
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
-16c=-144
\\\\
c=\dfrac{-144}{-16}
\\\\
c=9
.\end{array}
Hence, the given equation has $1$ rational solution when $
c=9
.$