Question

How can you solve this inequality using an algebraic method.
-x^3+2x^2+4x-8 > or equal to 0
(x-2)(-x+2)(x+2) > or equal to 0

Answer 1

you have correctly determined that
y = -(x-2)^2 (x+2)

You know the graph crosses the axis at x = -2 and that it just touches the x-axis at x=2 (because of the double root)

Since for large negative values of x, all 3 factors are negative,

y > 0 for x < -2

So, y < 0 for -2 < x < 2 because only one of the factors is negative

But y is also < 0 for x > 2 because all three factors are positive

So, y >= 0 for x <= -2

A nice graph can be seen at

http://www.wolframalpha.com/input/?i=-x^3%2B2x^2%2B4x-8