Answer
$x=-4$ or $x=\dfrac{2}{3}$
Work Step by Step
Using the factoring of trinomials in the form $ax^2+bx+c,$ the expression
\begin{align*}
3x^2+10x-8
\end{align*} has $ac=
3(-8)=-24
$ and $b=
10
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-2,12
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{align*}
3x^2-2x+12x-8=0
.\end{align*}
Grouping the first and second terms and the third and fourth terms, the expression above is equivalent to
\begin{align*}
(3x^2-2x)+(12x-8)=0
.\end{align*}
Factoring the $GCF$ in each group results to
\begin{align*}
x(3x-2)+4(3x-2)=0
.\end{align*}
Factoring the $GCF=
(3x-2)
$ of the entire expression above results to
\begin{align*}
(3x-2)(x+4)=0
.\end{align*}
Equating each factor to zero (Zero Product Property) results to \begin{align*} 3x-2&=0 \\\\\text{ OR }\\\\ x+4&=0 .\end{align*} Solving each equation above results to \begin{align*} 3x-2&=0 \\ 3x&=2 \\ \dfrac{3x}{3}&=\dfrac{2}{3} \\ x&=\dfrac{2}{3} \\\\\text{ OR }\\\\ x+4&=0 \\ x&=-4 .\end{align*} Hence, the solutions are
$x=-4$ or $x=\dfrac{2}{3}$.
