## Solving the Equation (x^2 + x)^2 = x^2 - 2x + 1

This equation looks complicated at first glance, but we can solve it by carefully expanding and simplifying. Here's how:

### 1. Expand the Left Side

First, we need to expand the left side of the equation:

(x² + x)² = (x² + x)(x² + x)

Using the distributive property (or FOIL method), we get:

(x² + x)² = x⁴ + 2x³ + x²

### 2. Rearrange the Equation

Now, let's rearrange the equation so that all terms are on one side:

x⁴ + 2x³ + x² - (x² - 2x + 1) = 0

Simplifying this, we get:

x⁴ + 2x³ + 2x - 1 = 0

### 3. Factorization

Unfortunately, this equation doesn't factor easily. There is no straightforward way to factor a fourth-degree polynomial like this. Therefore, we need to use more advanced techniques to find the solutions.

### 4. Numerical Solutions

To find the solutions, we can use numerical methods like the **Rational Root Theorem** or **Newton-Raphson method**. These methods involve iterative calculations to approximate the solutions.

**Rational Root Theorem:** This theorem helps us identify possible rational roots of the polynomial. We can use this information to test potential solutions and narrow down the search for the actual roots.

**Newton-Raphson Method:** This is an iterative method that starts with an initial guess for the root and then repeatedly refines the guess until it converges to the actual root.

### 5. Conclusion

Solving the equation (x² + x)² = x² - 2x + 1 involves expanding, rearranging, and utilizing numerical methods to find the solutions. While the equation doesn't factor easily, techniques like the Rational Root Theorem and Newton-Raphson method can help us find the approximate values of the roots.