Answer
$(-5, -3) U (5, ∞)$
Work Step by Step
$x^3+3x^2-25x-75 > 0$
$x^2(x+3)-25(x+3)>0$
$(x^2-25)(x+3)>0$
$(x-5)(x+5)(x+3)>0$
$x-5=0$
$x-5+5=0+5$
$x=5$
$x+5=0$
$x+5-5=0-5$
$x=-5$
$x+3=0$
$x+3-3=0-3$
$x=-3$
Four regions to test: $(-∞, -5)$, $(-5, -3)$, $(-3, 5)$, $(5, ∞)$
Let $x=-10$, $x=-4$, $x=4$, $x=10$
$x=-10$
$(x-5)(x+5)(x+3)>0$
$(-10-5)(-10+5)(-10+3)>0$
$-15*-5*-7 >0$
$75*-7 >0$
$-525 >0$ (false)
$x=-4$
$(x-5)(x+5)(x+3)>0$
$(-4-5)(-4+5)(-4+3)>0$
$-9*1*-1 >0$
$9 > 0$ (true)
$x=4$
$(x-5)(x+5)(x+3)>0$
$(4-5)(4+5)(4+3)>0$
$-1*9*7 >0$
$-63 >0$ (false)
$x=10$
$(x-5)(x+5)(x+3)>0$
$(10-5)(10+5)(10+3)>0$
$5*15*13 >0$
$5*195 >0$
$975 >0$ (true)
