## Solving the Equation: (x+1)^2-(x-2)^2=(x+3)^2+x^2-20

This article explores the process of solving the given equation: **(x+1)^2-(x-2)^2=(x+3)^2+x^2-20**. We will utilize algebraic manipulation to isolate the variable 'x' and find its solution.

### Expanding and Simplifying

Firstly, we expand the squares on both sides of the equation using the formula (a+b)^2 = a^2 + 2ab + b^2.

**(x+1)^2-(x-2)^2=(x+3)^2+x^2-20**

**x^2 + 2x + 1 - (x^2 - 4x + 4) = x^2 + 6x + 9 + x^2 - 20**

Next, we distribute the negative sign on the left side and combine like terms.

**x^2 + 2x + 1 - x^2 + 4x - 4 = 2x^2 + 6x - 11**

**6x - 3 = 2x^2 + 6x - 11**

### Isolating the Variable

Now, we rearrange the equation to set it equal to zero.

**0 = 2x^2 - 11 + 3**

**0 = 2x^2 - 8**

Next, we divide both sides by 2.

**0 = x^2 - 4**

Finally, we factor the equation.

**0 = (x + 2)(x - 2)**

### Finding the Solution

Setting each factor equal to zero, we solve for 'x'.

**x + 2 = 0** or **x - 2 = 0**

**x = -2** or **x = 2**

Therefore, the solutions to the equation **(x+1)^2-(x-2)^2=(x+3)^2+x^2-20** are **x = -2** and **x = 2**.