You have the rational expression:

$\frac{x^3 - 3x^2 - 8x - 4}{x + 1}$

To simplify this, we can use polynomial long division. Let's divide the cubic polynomial $x^3 - 3x^2 - 8x - 4$ by $x + 1$.

Steps for Polynomial Long Division:

1. Divide the first term of the numerator by the first term of the denominator: $\frac{x^3}{x} = x^2$ So, the first term in the quotient is $x^2$.

2. Multiply the entire divisor $x + 1$ by $x^2$: $x^2(x + 1) = x^3 + x^2$

3. Subtract $x^3 + x^2$ from the original polynomial: $(x^3 - 3x^2 - 8x - 4) - (x^3 + x^2) = -4x^2 - 8x - 4$

4. Divide the first term of the new polynomial $-4x^2$ by $x$: $\frac{-4x^2}{x} = -4x$ So, the next term in the quotient is $-4x$.

5. Multiply $-4x$ by $x + 1$: $-4x(x + 1) = -4x^2 - 4x$

6. Subtract $-4x^2 - 4x$ from $-4x^2 - 8x - 4$: $(-4x^2 - 8x - 4) - (-4x^2 - 4x) = -4x - 4$

7. Divide $-4x$ by $x$: $\frac{-4x}{x} = -4$ So, the next term in the quotient is $-4$.

8. Multiply $-4$ by $x + 1$: $-4(x + 1) = -4x - 4$

9. Subtract $-4x - 4$ from $-4x - 4$: $(-4x - 4) - (-4x - 4) = 0$

The remainder is 0, so the division is exact.

Final Result:

$\frac{x^3 - 3x^2 - 8x - 4}{x + 1} = x^2 - 4x - 4$

This is the simplified form of the given expression.

---

Do you want to explore further steps or have any questions about the process?

Here are 5 related questions you can explore:

1. What are other methods of dividing polynomials?
2. How do you check the result of a polynomial division?
3. What is synthetic division, and how is it different from long division?
4. Can this polynomial division be applied to higher-degree polynomials?
5. How can factoring help in simplifying rational expressions?

Tip: Always check the denominator for any restrictions, such as values that make it zero, to avoid undefined expressions.