DIVIDE X^6 -7^3 -8 BY THE PRODUCT OF X+1 AND ( X^2 +4) ?????

Answer 1

lose the CAPS LOCK, ok?

If, as I suspect, you meant

(x^6-7x^3-8) / (x+1)(x^2+2x+4)

note that x^6-7x^3-8 = (x^3-8)(x^3+1)
Now factor those two differences of cubes, ad you have

(x-2)(x^2+2x+4)(x+1)(x^2-x+1) / (x+1)(x^2+4)
= (x-2)(x^2-x+1)