Answer
$x=\left\{ -2,-\dfrac{6}{5},3 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
(x+2)(5x^2-9x-18)=0
,$ express it first in factored form. Then equate each factor to zero using the Zero Product Property. Finally, solve each resulting equation.
$\bf{\text{Solution Details:}}$
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
5(-18)=-90
$ and the value of $b$ is $
-9
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
6,-15
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
(x+2)(5x^2+6x-15x-18)=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}
(x+2)[(5x^2+6x)-(15x+18)]=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
(x+2)[x(5x+6)-3(5x+6)]=0
.\end{array}
Factoring the $GCF=
(5x+6)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(x+2)[(5x+6)(x-3)]=0
\\\\
(x+2)(5x+6)(x-3)=0
.\end{array}
Equating each factor to zero (Zero Product Property), the solutions to the equation above are
\begin{array}{l}\require{cancel}
x+2=0
\\\\\text{OR}\\\\
5x+6=0
\\\\\text{OR}\\\\
x-3=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x+2=0
\\\\
x=-2
\\\\\text{OR}\\\\
5x+6=0
\\\\
5x=-6
\\\\
x=-\dfrac{6}{5}
\\\\\text{OR}\\\\
x-3=0
\\\\
x=3
.\end{array}
Hence, $
x=\left\{ -2,-\dfrac{6}{5},3 \right\}
.$