## Simplifying Rational Expressions: (x^3+3x^2-4x-12)/(x^2+5x+6)

This article will explore the process of simplifying the rational expression **(x^3+3x^2-4x-12)/(x^2+5x+6)**.

### 1. Factor the numerator and denominator.

**Numerator:**- We can factor by grouping:
- (x^3 + 3x^2) + (-4x - 12)
- x^2(x + 3) - 4(x + 3)
**(x + 3)(x^2 - 4)**- We can further factor (x^2 - 4) as a difference of squares:
**(x + 3)(x + 2)(x - 2)**

**Denominator:**- We can factor the quadratic:
**(x + 2)(x + 3)**

### 2. Identify common factors

Now our expression looks like this: **[(x + 3)(x + 2)(x - 2)] / [(x + 2)(x + 3)]**
We can see that both the numerator and denominator share the factors (x + 3) and (x + 2).

### 3. Simplify by canceling common factors.

We can cancel out the common factors, leaving us with: **(x - 2) / 1**

### 4. Final Simplified Expression

The simplified form of the rational expression (x^3+3x^2-4x-12)/(x^2+5x+6) is **(x - 2)**.

**Important Note:** It's crucial to remember that the original expression and the simplified one are equivalent **except** when x = -3 or x = -2. These values make the original denominator zero, rendering the expression undefined. The simplified form doesn't reflect this restriction.