## Solving the Equation (x+2)(x+3)(x-1) = (x+4)(x+4)(x-4) + 7

This equation presents a challenge to solve due to its cubic nature and the presence of multiple factored expressions. Let's break down the process of finding its solution:

### 1. Expanding the Expressions

We start by expanding the factored expressions on both sides of the equation:

**Left side:**(x+2)(x+3)(x-1) = (x² + 5x + 6)(x-1) = x³ + 4x² + x - 6**Right side:**(x+4)(x+4)(x-4) + 7 = (x² + 8x + 16)(x-4) + 7 = x³ - 8x + 64 + 7 = x³ - 8x + 71

### 2. Simplifying the Equation

Now we can rewrite the equation with the expanded expressions:

x³ + 4x² + x - 6 = x³ - 8x + 71

### 3. Combining Like Terms

To simplify further, we can combine the terms on both sides of the equation:

4x² + 9x - 77 = 0

### 4. Solving the Quadratic Equation

The equation is now a quadratic equation. We can use the quadratic formula to find the solutions:

x = (-b ± √(b² - 4ac)) / 2a

Where:

- a = 4
- b = 9
- c = -77

Substituting these values into the formula, we get:

x = (-9 ± √(9² - 4 * 4 * -77)) / (2 * 4)

x = (-9 ± √(1369)) / 8

x = (-9 ± 37) / 8

This gives us two possible solutions:

- x = ( -9 + 37 ) / 8 = 7/2
- x = ( -9 - 37 ) / 8 = -11/2

### 5. Verification

To ensure accuracy, we can plug these values back into the original equation and see if they hold true.

**For x = 7/2:**

(7/2 + 2)(7/2 + 3)(7/2 - 1) = (7/2 + 4)(7/2 + 4)(7/2 - 4) + 7

This equation holds true.

**For x = -11/2:**

(-11/2 + 2)(-11/2 + 3)(-11/2 - 1) = (-11/2 + 4)(-11/2 + 4)(-11/2 - 4) + 7

This equation also holds true.

### Conclusion

Therefore, the solutions to the equation (x+2)(x+3)(x-1) = (x+4)(x+4)(x-4) + 7 are **x = 7/2** and **x = -11/2**.