## Solving the Equation: (x + 1/x)² + 2(x + 1/x) - 8 = 0

This equation might look intimidating at first, but we can solve it using a simple substitution and some basic algebraic manipulation.

### 1. Substitute for Simplicity

Let's make the equation easier to work with by substituting:

**y = x + 1/x**

Now our equation becomes:

**y² + 2y - 8 = 0**

### 2. Factor the Quadratic Equation

This is a standard quadratic equation, which we can factor:

**(y + 4)(y - 2) = 0**

This gives us two possible solutions for y:

**y = -4****y = 2**

### 3. Substitute Back and Solve for x

Now, let's substitute back our original expression for y:

**Case 1: y = -4**

x + 1/x = -4

Multiplying both sides by x:

x² + 1 = -4x

Rearranging:

x² + 4x + 1 = 0

This is a quadratic equation, which we can solve using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = 1, b = 4, and c = 1

Solving this gives us two solutions for x:

**x = -2 + √3**
**x = -2 - √3**

**Case 2: y = 2**

x + 1/x = 2

Multiplying both sides by x:

x² + 1 = 2x

Rearranging:

x² - 2x + 1 = 0

This is a perfect square trinomial:

(x - 1)² = 0

Therefore, the solution for this case is:

**x = 1**

### 4. Conclusion

Therefore, the solutions for the equation (x + 1/x)² + 2(x + 1/x) - 8 = 0 are:

**x = -2 + √3****x = -2 - √3****x = 1**