Answer
$h = 0, -2±2\sqrt {3}$
Work Step by Step
$4h^{3} + 16h^{2} - 32h = 0$
$4h(h^{2} + 4h - 8) = 0$
$h = \frac{-(4)±\sqrt {(4)^{2}-4(1)(-8)}}{2(1)}$
$h = \frac{-4±\sqrt {16-4(1)(-8)}}{2(1)}$
$h = \frac{-4±\sqrt {16+32}}{2}$
$h = \frac{-4±\sqrt {48}}{2}$
$h = \frac{-4±2\sqrt {12}}{2}$
$h = \frac{-2±\sqrt {12}}{1}$
$h = 0, -2±\sqrt {12}$
$h = 0, -2±2\sqrt {3}$
Check:
When $h = 0$
$4(0)^{3} + 16(0)^{2} - 32(0) \overset{?}{=} 0$
$0 + 0 - 0 \overset{?}{=}0$
$0 = 0$
When $h = -2+2\sqrt {3}$
$4(-2+2\sqrt {3})^{3} + 16(-2+2\sqrt {3})^{2} - 32(-2+2\sqrt {3}) \overset{?}{=} 0$
$4(3.1384...) + 16(2.14359...) - (46.851...)\overset{?}{=} 0$
$(46.851...) - (46.851...)\overset{?}{=} 0$
$0 = 0$
When $h = -2-2\sqrt {3}$
$4(-2-2\sqrt {3})^{3} + 16(-2-2\sqrt {3})^{2} - 32(-2-2\sqrt {3}) \overset{?}{=} 0$
$4(-163.1384...) + 16(29.856...) - (-174.8512...)\overset{?}{=} 0$
$(-652.5537...) + (477.702...) + (174.8512...)\overset{?}{=} 0$
$(-652.5537...) + (-652.5537...) \overset{?}{=} 0$
$0 = 0$