To solve the expression:

$\frac{x^3 + x^2 - 4x - 4}{x^2 + 7x - 18}$

we can follow these steps:

Step 1: Factor the denominator

The denominator is $x^2 + 7x - 18$. We can factor this quadratic expression by finding two numbers that multiply to $-18$ and add up to $7$.

The numbers are $9$ and $-2$, so we factor it as:

$x^2 + 7x - 18 = (x + 9)(x - 2)$

Step 2: Try factoring the numerator

The numerator is $x^3 + x^2 - 4x - 4$. Let's try factoring this cubic expression by grouping:

Group the terms:

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

Now, factor each group:

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

We can factor out the common factor $(x + 1)$:

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

Now, notice that $x^2 - 4$ is a difference of squares, which can be factored as:

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

So the numerator becomes:

$(x + 1)(x - 2)(x + 2)$

Step 3: Simplify the expression

Now we have the factored form of the expression:

$\frac{(x + 1)(x - 2)(x + 2)}{(x + 9)(x - 2)}$

Since $(x - 2)$ appears in both the numerator and denominator, we can cancel it out:

$\frac{(x + 1)(x + 2)}{(x + 9)}$

Final Answer:

$\frac{(x + 1)(x + 2)}{(x + 9)}$

This is the simplified form of the expression.

Do you have any questions or want further details? Here are five related questions to deepen your understanding:

1. How do you factor quadratic expressions like $x^2 + 7x - 18$?
2. What is the difference of squares, and how does it apply to $x^2 - 4$?
3. How do you identify common factors in polynomials?
4. Why can we cancel out factors like $x - 2$ in rational expressions?
5. What are some other methods for simplifying rational expressions?

Tip: Always check if terms can be factored and canceled to simplify a complex rational expression.