Question

create a quintic polynomial inequality for which x=-4, x=0, x ≥ 2 is the solution. Justify your answer by solving the inequality using an interval chart. Provide a diagram as well.

Answer 1

Your wording is kinda murky. It appears you want the graph to cross the x-axis at (-4,0), (0,0) and (2,0), and stay above the axis for x≥2. I'd guess you want y≥0 on the interval [-4,0]U[2,∞).
So you must want y≥0 for some quintic polynomial.
Start with y = (x+4)(x)(x-2)(ax^2+bx+c)
As long as the discriminant of the quadratic is negative, there will be no other real roots, so make it easy.
y = x(x+4)(x-2)(x^2+1)

This graph is at

https://www.wolframalpha.com/input/?i=x%28x%2B4%29%28x-2%29%28x%5E2%2B1%29+for+-5+%3C%3Dx+%3C%3D+3

If I have misread the problem somehow, please fix it up so we can get it right.