What is the quotient of (x^3 + 8) divided by (x + 2)?

1. Identify the problem: The problem asks for the quotient of the polynomial $x^3 + 8$ divided by the binomial $x + 2$. This is a polynomial division problem.

1. Recognize the sum of cubes: The expression $x^3 + 8$ can be recognized as a sum of cubes, since $8 = 2^3$. The formula for the sum of cubes is $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$.

3. Apply the sum of cubes formula: In this case, $a = x$ and $b = 2$. Applying the formula, we get:
$$x^3 + 8 = (x+2)(x^2 - x(2) + 2^2)$$
$$x^3 + 8 = (x+2)(x^2 - 2x + 4)$$

4. Perform the division: We need to find the quotient of $(x^3 + 8) \div (x + 2)$. Using the factored form from the previous step:
$$\frac{x^3 + 8}{x + 2} = \frac{(x+2)(x^2 - 2x + 4)}{x + 2}$$

5. Simplify the expression: Cancel out the common factor $(x+2)$ from the numerator and the denominator:
$$\frac{(x+2)(x^2 - 2x + 4)}{x + 2} = x^2 - 2x + 4$$

Alternatively, we can use polynomial long division or synthetic division. Using synthetic division with the root of the divisor $x+2=0$, which is $x=-2$:

-2 | 1 0 0 8
| -2 4 -8
----------------
1 -2 4 0

The resulting coefficients are 1, -2, and 4, which correspond to the polynomial $1x^2 - 2x + 4$. The remainder is 0.

1. State the final answer: The quotient is $x^2 - 2x + 4$.