Answer
$(-∞, -6) U (-3/4, 0) U (5, ∞)$
Work Step by Step
$(4x+3)(x-5)/(x)(x+6) > 0$
$4x+3=0$
$4x+3-3=0-3$
$4x=-3$
$4x/4=-3/4$
$x=-3/4$
$x-5=0$
$x-5+5=0+5$
$x=5$
$x=0$
$x+6=0$
$x+6-6=0-6$
$x=-6$
Five regions to test: $(-∞, -6)$, $(-6, -3/4)$, $(-3/4, 0)$, $(0, 5)$, $(5, ∞)$
Let $x=-10$, $x=-2$, $x=-1/2$, $x=2$, $x=10$
$x=-10$
$(4x+3)(x-5)/(x)(x+6) > 0$
$(4*-10+3)(-10-5)/(-10)(-10+6) > 0$
$(-40+3)(-15)/(-10)(-4) > 0$
$(-37*-15)/40 > 0$
$555/40 > 0$
$13.875 > 0$ (true)
$x=-2$
$(4x+3)(x-5)/(x)(x+6) > 0$
$(4*-2+3)(-2-5)/(-2)(-2+6) > 0$
$(-12+3)(-7)/(-2)(4) > 0$
$(-9)(-7)/(-8) > 0$
$63/-8 > 0$ (false)
$x=-1/2$
$(4x+3)(x-5)/(x)(x+6) > 0$
$(4*-1/2+3)(-1/2-5)/(-1/2)(-1/2+6) > 0$
$(-2+3)(-11/2)/(-1/2)((11/2) >0$
$(1)(-11/2)/(-11/4) >0$
$-11/2 / -11/4 > 0$
$-5.5 / -2.75 >0$
$2 > 0$ (true)
$x=2$
$(4x+3)(x-5)/(x)(x+6) > 0$
$(4*2+3)(2-5)/(2)(2+6) > 0$
$(8+3)(-3)/(2)(8) > 0$
$11*-3/16 > 0$
$-33/16 > 0$ (false)
$x=10$
$(4x+3)(x-5)/(x)(x+6) > 0$
$(4*10+3)(10-5)/(10)(10+6) > 0$
$(40+3)(5)/(10)(16) >0$
$(43)/(2*16) >0$
$43/32 > 0$ (true)
