Answer
$x=\left\{ -2,-\dfrac{2}{3} \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given eqution, $
3x^2+8x=-4
,$ express the equation in the form $ax^2+bx+c=0.$ Then express the equation in factored form. Next step is to equate each factor to zero (Zero Product Property). Finally, solve each equation.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
3x^2+8x+4=0
.\end{array}
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
3(4)=12
$ and the value of $b$ is $
8
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
2,6
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
3x^2+2x+6x+4=0
.\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}
(3x^2+2x)+(6x+4)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
x(3x+2)+2(3x+2)=0
.\end{array}
Factoring the $GCF=
(3x+2)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(3x+2)(x+2)=0
.\end{array}
Equating each factor to zero (Zero Product Property), the solutions to the equation above are
\begin{array}{l}\require{cancel}
3x+2=0
\\\\\text{OR}\\\\
x+2=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3x+2=0
\\\\
3x=-2
\\\\
x=-\dfrac{2}{3}
\\\\\text{OR}\\\\
x+2=0
\\\\
x=-2
.\end{array}
Hence, $
x=\left\{ -2,-\dfrac{2}{3} \right\}
.$
