You want a product to be negative. A product is negative if and only if the two factors have opposite signs.
So, there are two cases: either [tex]x-3[/tex] is positive and [tex]x+5[/tex] is negative, or vice versa. But since [tex]x-3[/tex] is less than [tex]x+5[/tex], the only possibility is that
[tex]\begin{cases}x-3<0\\x+5>0\end{cases} \iff \begin{cases}x<3\\x>-5\end{cases}[/tex]
So, the solution is given by all numbers [tex]x[/tex] such that
[tex]-5<x<3[/tex]
