Answer
$[-2]$ U $[2,∞)$
Work Step by Step
$x^3+2x^2-4x\geq8$
$x^3+2x^2-4x=8$
$x^3+2x^2-4x-8=8-8$
$x^3+2x^2-4x-8=0$
$x^2(x+2)-4(x+2)=0$
$(x^2-4)(x+2)=0$
$x^2-4=0$
$x^2-4+4=0+4$
$x^2=4$
$\sqrt {x^2} = \sqrt 4$
$x=±2$
$x+2=0$
$x+2-2=0-2$
$x=-2$
We have the four regions to test: $(-∞, -2]$, $[-2]$, $[-2,2]$, $[2,∞)$
Let $x=-3$, $x=-2$, $x=0$, $x=3$
$x=-3$
$x^3+2x^2-4x\geq8$
$(-3)^3+2*(-3)^2-4*(-3)\geq8$
$-27+2*9+12 \geq 8$
$-27 +18+12 \geq 8$
$-27+30 \geq 8$
$-3 \geq 8$ (false)
$x=-2$
$x^3+2x^2-4x\geq8$
$(-2)^3+2*(-2)^2-4*(-2)\geq8$
$-8 +2*4+8 \geq 8$
$8 \geq 8$ (true)
$x=0$
$x^3+2x^2-4x\geq8$
$0^3+2*0^2-4*0\geq8$
$0+2*0-0 \geq 8$
$0+0-0 \geq 8$
$0 \geq 8$ (false)
$x=3$
$x^3+2x^2-4x\geq8$
$3^3+2*3^2-4*3\geq8$
$27+2*9-12 \geq 8$
$27+18-12 \geq 8$
$33 \geq 8$ (true)
