## Solving the Equation: (x+1/x+2)2 = x+2/x+4

This equation presents a unique challenge due to the presence of fractions and squares. Let's break down the steps to find the solution:

### 1. Simplify the Equation

Begin by simplifying the left side of the equation. Expand the square:

(x+1/x+2)² = (x+1/x+2) * (x+1/x+2)

Using the FOIL method (First, Outer, Inner, Last):

= x² + (2x/x+2) + (x/x+2) + 1/(x+2)²

Combining like terms:

= x² + (3x/x+2) + 1/(x+2)²

Now the equation becomes:

x² + (3x/x+2) + 1/(x+2)² = x+2/x+4

### 2. Eliminate Fractions

To get rid of the fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is (x+2)²(x+4):

(x+2)²(x+4) * [x² + (3x/x+2) + 1/(x+2)²] = (x+2)²(x+4) * (x+2/x+4)

This simplifies to:

(x+2)²(x+4)x² + 3x(x+2)(x+4) + (x+4) = (x+2)³

### 3. Expand and Rearrange

Expand the terms on both sides of the equation and move all terms to one side:

x⁵ + 8x⁴ + 20x³ + 16x² - 8x - 8 = 0

### 4. Solve for x

The resulting equation is a quintic equation, which generally does not have a simple analytical solution. Therefore, we would need to resort to numerical methods to find the approximate values of x. These methods include:

**Graphical Solution:**Plotting the function f(x) = x⁵ + 8x⁴ + 20x³ + 16x² - 8x - 8 and identifying the x-intercepts.**Numerical Methods:**Using iterative algorithms like Newton-Raphson method to approximate the roots.

**Important Note:** It is crucial to check the validity of the solutions obtained from these methods by substituting them back into the original equation to ensure they satisfy the equation.

### Conclusion

While a closed-form solution to the equation (x+1/x+2)² = x+2/x+4 is not readily available, we can use numerical methods to approximate the solutions. Remember to check the validity of the solutions obtained.