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 $x \neq 2$.

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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.