This is a division of polynomials. It is supposed to be divided by using the division algorithm. Please help me!

There are a couple of ways to attack this, but I will choose my way.

Let's rewrite the expression as:

(x^3 - 8)/(x - 2) + 1/(x - 2)

This is equal to our original expression.

When factoring a difference of two cubes you get this:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

x^3 - 8 = (x - 2)(x^2 +2x + 4)

Now we have:

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

You can cancel out the x-2 term in the numerator & denominator. You get:

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