## Solving the Equation: (x+2)^2 - 3x - 5 = (1-x)(1+x)

This article will guide you through the steps to solve the equation (x+2)^2 - 3x - 5 = (1-x)(1+x). We will employ algebraic manipulation and simplification to determine the possible values for 'x' that satisfy the given equation.

### Step 1: Expanding both sides of the equation

**Left-hand side:**- Expand (x+2)^2: (x+2)^2 = x^2 + 4x + 4
- Now the left side becomes: x^2 + 4x + 4 - 3x - 5

**Right-hand side:**- Expand (1-x)(1+x): (1-x)(1+x) = 1 - x^2
- This is a common pattern known as the "difference of squares".

Now the equation looks like this: x^2 + 4x + 4 - 3x - 5 = 1 - x^2

### Step 2: Simplifying the equation

- Combine like terms on both sides: 2x^2 + x - 2 = 0

### Step 3: Solving the quadratic equation

We now have a quadratic equation in the form ax^2 + bx + c = 0. There are a few ways to solve this:

**Factoring:**Try to find two numbers that add up to 'b' (1) and multiply to 'ac' (2 * -2 = -4). In this case, the numbers are 2 and -1. We can then rewrite the equation as: (2x - 1)(x + 2) = 0. This gives us two possible solutions:- 2x - 1 = 0 --> x = 1/2
- x + 2 = 0 --> x = -2

**Quadratic formula:**This formula works for any quadratic equation:- x = (-b ± √(b^2 - 4ac)) / 2a
- Substitute the values from our equation: x = (-1 ± √(1^2 - 4 * 2 * -2)) / (2 * 2)
- Simplify: x = (-1 ± √17) / 4
- This gives us two solutions:
- x = (-1 + √17) / 4
- x = (-1 - √17) / 4

### Conclusion

The solutions to the equation (x+2)^2 - 3x - 5 = (1-x)(1+x) are:

**x = 1/2****x = -2****x = (-1 + √17) / 4****x = (-1 - √17) / 4**