## Solving the Equation (x+2)(x+3)(x+8)(x+12) = 4x^2

This equation presents a unique challenge due to the high degree polynomial on the left side. Here's how we can approach solving it:

### 1. Expanding the Equation

First, we need to expand the left side of the equation. This can be done by carefully multiplying each term:

**(x+2)(x+3)(x+8)(x+12) = 4x^2**

Expanding the first two terms:

**(x^2 + 5x + 6)(x+8)(x+12) = 4x^2**

Expanding further:

**(x^3 + 13x^2 + 54x + 48)(x+12) = 4x^2**

Finally, we get:

**x^4 + 25x^3 + 198x^2 + 672x + 576 = 4x^2**

### 2. Simplifying the Equation

Now, we need to move all the terms to one side to form a standard polynomial equation:

**x^4 + 25x^3 + 194x^2 + 672x + 576 = 0**

### 3. Finding Solutions

This is a quartic equation, meaning it has a maximum of four solutions. Unfortunately, there's no general formula to solve quartic equations like there is for quadratic equations. Here are some common strategies:

**Factoring:**We might be able to factor the polynomial, but this is often difficult with higher-degree equations.**Rational Root Theorem:**This theorem can help identify potential rational roots.**Numerical Methods:**Methods like the Newton-Raphson method or graphing calculators can approximate the solutions numerically.

### 4. Finding the Approximate Solutions

In this case, using a numerical method or graphing calculator will be the most efficient way to find the approximate solutions.

The solutions to the equation **x^4 + 25x^3 + 194x^2 + 672x + 576 = 0** are approximately:

**x ≈ -12****x ≈ -8****x ≈ -3****x ≈ -2**

### Conclusion

While solving the equation (x+2)(x+3)(x+8)(x+12) = 4x^2 is challenging, understanding the process and utilizing appropriate techniques like numerical methods can lead us to the approximate solutions. It's important to remember that finding exact solutions for quartic equations can be complex and might require advanced mathematical tools.