## Solving the Equation: (x+1)^2(x+2) + (x-1)^2(x-2) = 12

This equation presents a challenge due to the presence of squared terms and multiple factors. Let's break it down step-by-step to find the solution.

### Expanding the Equation

First, we need to expand the equation by multiplying out the terms:

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

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

Now the equation becomes:

**(x^2 + 2x + 1)(x+2) + (x^2 - 2x + 1)(x-2) = 12**

Let's multiply further:

**(x^2 + 2x + 1)(x+2) = x^3 + 4x^2 + 5x + 2****(x^2 - 2x + 1)(x-2) = x^3 - 4x^2 + 5x - 2**

The equation is now:

**x^3 + 4x^2 + 5x + 2 + x^3 - 4x^2 + 5x - 2 = 12**

### Simplifying the Equation

Combining like terms, we get:

**2x^3 + 10x = 12**

Let's move all terms to one side:

**2x^3 + 10x - 12 = 0**

We can simplify this by dividing all terms by 2:

**x^3 + 5x - 6 = 0**

### Finding the Solution

Now, we have a cubic equation. Finding the exact solution for cubic equations can be complex. There are multiple methods, including factoring, using the Rational Root Theorem, and numerical methods.

**Here's how we can approach finding the solution:**

**Factorization:**We can try to factor the equation. However, it doesn't appear to be easily factorable.**Rational Root Theorem:**This theorem helps us find potential rational roots. Unfortunately, this method doesn't guarantee a solution, but it can give us possibilities to test.**Numerical Methods:**Methods like the Newton-Raphson method can be used to approximate the solution numerically.

In this case, we can observe that the equation has one real root, and it lies between 1 and 2. We can use numerical methods to get a more accurate value for the solution.

### Conclusion

The equation (x+1)^2(x+2) + (x-1)^2(x-2) = 12 has one real solution. While we can't find an exact analytical solution, we can use numerical methods to approximate the solution.