## Solving the Equation: (x+3)(x+1)(x-7) = (x+3)(x+1)(x-8)

This equation presents a unique situation where we can solve for the value of *x* without expanding the entire expressions. Let's break down the solution process:

### 1. Recognizing the Common Factors

Observe that both sides of the equation share the same factors: (x+3) and (x+1). This allows us to simplify the equation significantly.

### 2. Simplifying the Equation

Divide both sides of the equation by the common factors (x+3)(x+1). This leaves us with:

(x-7) = (x-8)

**Important Note:** This step is only valid if (x+3) and (x+1) are not equal to zero. We'll address this later.

### 3. Solving for x

The simplified equation, (x-7) = (x-8), leads to a contradiction. Subtracting *x* from both sides, we get -7 = -8, which is impossible.

### 4. Analyzing the Solution

The contradiction indicates that there is **no solution** for the original equation. This is because the factors (x+3) and (x+1) must be equal to zero for the equation to hold true. However, if either of these factors is zero, it would make the entire expression equal to zero on both sides, regardless of the value of the other factor.

### 5. The Solution Set

Therefore, the solution set for the equation (x+3)(x+1)(x-7) = (x+3)(x+1)(x-8) is **empty**, meaning there is no value of *x* that satisfies the equation.