To simplify the expression \(\frac{4 - x^2}{x^2 + x - 6}\), we'll start by factoring both the numerator and the denominator.

### Step 1: Factor the Numerator
The numerator is \(4 - x^2\). This can be written as a difference of squares:

\[
4 - x^2 = (2)^2 - (x)^2 = (2 - x)(2 + x)
\]

### Step 2: Factor the Denominator
The denominator is \(x^2 + x - 6\). We need to find two numbers that multiply to \(-6\) and add to \(1\) (the coefficient of \(x\)).

These numbers are \(3\) and \(-2\). So, we can factor the denominator as:

\[
x^2 + x - 6 = (x + 3)(x - 2)
\]

### Step 3: Rewrite the Expression with the Factored Form
Substitute the factored forms into the expression:

\[
\frac{4 - x^2}{x^2 + x - 6} = \frac{(2 - x)(2 + x)}{(x + 3)(x - 2)}
\]

### Step 4: Simplify the Expression
Notice that \((2 - x)\) and \((x - 2)\) are negatives of each other:

\[
2 - x = -(x - 2)
\]

Thus, we can rewrite the expression:

\[
\frac{(2 - x)(2 + x)}{(x + 3)(x - 2)} = \frac{-(x - 2)(2 + x)}{(x + 3)(x - 2)}
\]

Now, cancel out the common terms \((x - 2)\) from the numerator and the denominator:

\[
= \frac{-(2 + x)}{x + 3}
\]

### Step 5: Simplify Further
Rewrite the simplified expression:

\[
= \frac{-1(2 + x)}{x + 3} = \frac{-(x + 2)}{x + 3}
\]

### Final Simplified Form
The simplified expression is:

\[
\frac{-(x + 2)}{x + 3}
\]

or equivalently:

\[
-\frac{x + 2}{x + 3}
\]

### Note on Domain Restrictions
The original expression has restrictions where the denominator is zero, so \(x \neq -3\) and \(x \neq 2\).

---

Would you like any more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you factor quadratic expressions in general?
2. Why is it important to check for restrictions on the variable when simplifying rational expressions?
3. What is the difference of squares formula and how is it applied?
4. How can we simplify expressions with higher-degree polynomials in the numerator and denominator?
5. What are the common mistakes to avoid when simplifying rational expressions?

**Tip:** Always check for common factors that can be canceled out, but remember to consider the domain restrictions before simplifying.

Simplify the rational expression (4 - x^2)/(x^2 + x - 6) with a detailed, step-by-step solution. This problem demonstrates algebraic techniques like factoring, simplifying rational expressions, and understanding the difference of squares. Ideal for students in Grades 8-10.

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