## Solving the Equation: (x+1/x-2)^2 + x+1/x-4 - 3(2x-4/x-4)^2 = 0

This equation presents a challenge due to its complex structure. We will break down the steps to solve it:

### 1. Simplification

**a. Factor out common terms:**

Observe that both the first and third terms have a common factor of (x-4) in the denominator. We can simplify the equation by factoring this out:

(x+1/x-2)^2 + (x+1)/(x-4) - 3(2x-4/(x-4))^2 = 0

**(x+1/x-2)^2 + (x+1)/(x-4) - 3(2(x-2)/(x-4))^2 = 0**

**b. Simplify the expressions:**

Now, we can further simplify by expanding the squares:

[(x+1)^2/(x-2)^2] + (x+1)/(x-4) - 3[4(x-2)^2/(x-4)^2] = 0

**c. Combine terms with the same denominator:**

To combine the terms, we need to find a common denominator. The common denominator for all terms is (x-2)^2(x-4)^2.

[(x+1)^2(x-4)^2 + (x+1)(x-2)^2(x-4) - 12(x-2)^2(x-4)] / [(x-2)^2(x-4)^2] = 0

### 2. Solve the Numerator

Since the denominator cannot be zero, we only need to solve the numerator:

(x+1)^2(x-4)^2 + (x+1)(x-2)^2(x-4) - 12(x-2)^2(x-4) = 0

**a. Factor out common terms:**

Notice that all terms have a common factor of (x-2)^2(x-4). We can factor this out:

(x-2)^2(x-4)[(x+1)^2 + (x+1)(x-2) - 12] = 0

**b. Simplify the expression inside the brackets:**

Expanding the expressions inside the brackets, we get:

(x-2)^2(x-4)[x^2 + 2x + 1 + x^2 - x - 2 - 12] = 0

**(x-2)^2(x-4)[2x^2 + x - 13] = 0**

### 3. Solve for x

Now we have a simpler equation to solve. We need to find the values of x that make this equation true:

**x - 2 = 0**=>**x = 2****x - 4 = 0**=>**x = 4****2x^2 + x - 13 = 0**

To solve the quadratic equation 2x^2 + x - 13 = 0, we can use the quadratic formula:

x = [-b ± √(b^2 - 4ac)] / 2a

Where a = 2, b = 1, and c = -13.

Solving for x using the quadratic formula, we get two more solutions:

**x = (-1 + √105) / 4****x = (-1 - √105) / 4**

### 4. Verification

It's important to verify if the solutions we found are valid. We need to check if any of the solutions make the original denominator zero.

We find that **x = 2 and x = 4** make the denominator zero, so these solutions are **extraneous** and need to be discarded.

### Conclusion

Therefore, the solutions to the equation (x+1/x-2)^2 + x+1/x-4 - 3(2x-4/x-4)^2 = 0 are:

**x = (-1 + √105) / 4****x = (-1 - √105) / 4**