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.

Learn how to simplify the rational expression x^3 + x^2 - 4x - 4 / x^2 + 7x - 18 through factoring. This solution covers factoring quadratics, cubic expressions, and the difference of squares theorem. Ideal for students in Grades 10-12.

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