Solving the Equation: (x + 1/x)^2 + 2(x + 1/x)  8 = 0
This equation appears complex at first glance, but we can simplify it using a clever substitution. Let's break down the steps:
1. Substitution:
Let's introduce a new variable, say y, to represent the expression (x + 1/x):
 y = (x + 1/x)
Now our equation becomes much simpler:
 y² + 2y  8 = 0
2. Solving the Quadratic Equation:
We now have a standard quadratic equation. We can solve it using the quadratic formula:
 y = (b ± √(b²  4ac)) / 2a
Where a = 1, b = 2, and c = 8.

y = (2 ± √(2²  4 * 1 * 8)) / (2 * 1)

y = (2 ± √(36)) / 2

y = (2 ± 6) / 2
This gives us two possible solutions for y:
 y1 = 2
 y2 = 4
3. Finding the Solutions for x:
Now we need to substitute back the original expression for y:

For y1 = 2:
 (x + 1/x) = 2
 x² + 1 = 2x
 x²  2x + 1 = 0
 (x  1)² = 0
 x = 1

For y2 = 4:
 (x + 1/x) = 4
 x² + 1 = 4x
 x² + 4x + 1 = 0
 Using the quadratic formula again, we get:
 x = (4 ± √(4²  4 * 1 * 1)) / (2 * 1)
 x = (4 ± √12) / 2
 x = (4 ± 2√3) / 2
 x = 2 ± √3
Therefore, the solutions for the equation (x + 1/x)² + 2(x + 1/x)  8 = 0 are:
 x = 1
 x = 2 + √3
 x = 2  √3