The given mathematical expression is:

$\frac{x^3 + x^2 - 4x - 4}{x + 1}$

We are asked to find the quotient of this division. Let's perform the division using polynomial long division.

Step-by-Step Polynomial Long Division:

1. Divide the first term of the numerator by the first term of the denominator:

   $\frac{x^3}{x} = x^2$

   Multiply $x^2$ by $x + 1$ and subtract:

   $(x^2)(x + 1) = x^3 + x^2$ Subtract from the original polynomial:

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

2. Divide the first term of the new polynomial by the first term of the denominator:

   $\frac{-4x}{x} = -4$

   Multiply $-4$ by $x + 1$ and subtract:

   $(-4)(x + 1) = -4x - 4$ Subtract from the new polynomial:

   $(-4x - 4) - (-4x - 4) = 0$

Thus, the quotient is:

$x^2 - 4$

The correct answer is:

$\boxed{x^2 - 4}$

Do you need further clarification on any step?

Here are 5 related questions for further practice:

1. How do you factor the polynomial $x^2 - 4$?
2. What is synthetic division and how does it differ from polynomial long division?
3. How can you verify the result of a division using multiplication?
4. What are the roots of the equation $x^2 - 4 = 0$?
5. How would you simplify the expression $\frac{x^3 - 4x}{x}$?

Tip: Always double-check your division results by multiplying the quotient by the divisor and comparing it to the original polynomial.