Answer
$x=\left\{ -\dfrac{2}{5},1 \right\}$
Work Step by Step
The two numbers whose product is $ac=
5(-2)=-10
$ and whose sum is $b=
-3
$ are $\{
-5,2
\}$. Using these two numbers to decompose the middle term of the given equation, $
5x^2-3x-2=0
,$ results to
\begin{array}{l}\require{cancel}
5x^2-5x+2x-2=0
\\\\
(5x^2-5x)+(2x-2)=0
\\\\
5x(x-1)+2(x-1)=0
\\\\
(x-1)(5x+2)=0
.\end{array}
Equating each factor to zero (or the Zero-Factor Property), the solutions to the given equation are
\begin{array}{l}\require{cancel}
x-1=0
\\\\
x=1
,\\\\\text{OR}\\\\
5x+2=0
\\\\
5x=-2
\\\\
x=-\dfrac{2}{5}
.\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{2}{5},1 \right\}
.$
