#### Answer

$x=-\dfrac{4}{3}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
Express the given equation, $
9x^2+24x+16=0
,$ in factored form. Then, use the Zero-Factor Property by equating each factor to zero. Finally, solve each equation.
$\bf{\text{Solution Details:}}$
The factored form of the equation above is
\begin{array}{l}\require{cancel}
(3x+4)(3x+4)=0
.\end{array}
Equating each factor to zero (Zero-Factor Property), then
\begin{array}{l}\require{cancel}
3x+4=0
\text{ OR }
3x+4=0
.\end{array}
Using the properties of equality to solve each of the equation above results to
\begin{array}{l}\require{cancel}
3x+4=0
\\\\
3x=-4
\\\\
x=-\dfrac{4}{3}
\\\\\text{ OR }\\\\
3x+4=0
\\\\
3x=-4
\\\\
x=-\dfrac{4}{3}
.\end{array}
Hence, the solutions are $
x=-\dfrac{4}{3}
.$
$\bf{\text{Supplementary Solution/s:}}$
To factor the quadratic expression $ax^2+bx+c,$ find two numbers whose product is $ac$ and whose sum is $b$. Use these $2$ numbers to decompose the middle term of the quadratic expression and then use factoring by grouping.
In the expression, $
9x^2+24x+16
,$ the value of $ac$ is $
9(16)=144
$ and the value of $b$ is $
24
.$
The $2$ numbers that have a product $ac$ and a sum of $b$ are $\{
12,12
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
9x^2+12x+12x+16
.\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}
(9x^2+12x)+(12x+16)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3x(3x+4)+4(3x+4)
.\end{array}
Factoring the $GCF=
(3x+4)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(3x+4)(3x+4)
.\end{array}