## Solving the Equation (x+1)(x+3)(x-5)(x-7) = 192

This problem involves solving a quartic equation (an equation with the highest power of the variable being 4). Let's break down the steps to find the solutions:

### 1. Expand the Equation

First, we need to expand the left-hand side of the equation by multiplying the factors:

```
(x+1)(x+3)(x-5)(x-7) = 192
```

Expanding the first two terms and the last two terms:

```
(x^2 + 4x + 3)(x^2 - 12x + 35) = 192
```

Now, we multiply these two quadratic expressions:

```
x^4 - 8x^3 - 67x^2 + 268x + 105 = 192
```

### 2. Rearrange the Equation

Bring all terms to one side to get a standard quartic equation:

```
x^4 - 8x^3 - 67x^2 + 268x - 87 = 0
```

### 3. Find the Solutions

Unfortunately, there's no general formula to directly solve quartic equations like there is for quadratic equations. We'll need to use other methods. Here are some options:

**Factoring:**Try to factor the quartic expression into simpler factors. In this case, it might be difficult to factor directly.**Rational Root Theorem:**This theorem helps identify potential rational roots of the equation. It states that any rational root of the equation must be of the form p/q, where p is a factor of the constant term (-87) and q is a factor of the leading coefficient (1). However, this method might not always provide all the solutions.**Numerical Methods:**Methods like the Newton-Raphson method or bisection method can be used to find approximate solutions numerically.**Graphing Calculator:**You can graph the function y = x^4 - 8x^3 - 67x^2 + 268x - 87 and find the points where the graph intersects the x-axis. These points represent the real solutions to the equation.

### 4. Finding Solutions (Example)

Let's try using the **Rational Root Theorem** to find potential solutions:

- Factors of -87: ±1, ±3, ±29, ±87
- Factors of 1: ±1

Therefore, the possible rational roots are ±1, ±3, ±29, ±87.

We can test these values by plugging them into the equation to see if they make the equation true.

**Important Note:** Finding the solutions for this specific equation might be quite challenging, and it's possible that the solutions involve complex numbers.

### Conclusion

Solving the equation (x+1)(x+3)(x-5)(x-7) = 192 involves expanding the equation, rearranging it into a standard quartic equation, and then using various methods to find the solutions. While there's no simple formula for quartic equations, tools like the Rational Root Theorem and numerical methods can help in finding the solutions.