## Solving the Equation (x+1)(x+3)(x+5)(x+7) = 5760

This problem involves solving a quartic equation, which can be quite challenging. However, we can use some clever techniques to simplify it and find the solution(s).

### Recognizing the Pattern and Simplification

Notice the pattern in the factors: (x+1), (x+3), (x+5), (x+7). They are all consecutive odd numbers. This pattern suggests we can try to manipulate the equation to make it easier to solve.

**Step 1: Manipulate the factors**

Let's introduce a new variable, `y = x + 3`

. This allows us to rewrite the equation:

```
(x + 1)(x + 3)(x + 5)(x + 7) = 5760
(y - 2)(y)(y + 2)(y + 4) = 5760
```

**Step 2: Simplify the expression**

Now, we can group the terms:

```
(y^2 - 4)(y^2 + 4y) = 5760
```

**Step 3: Expand and Rearrange**

Expand the expression and move all terms to one side:

```
y^4 + 4y^3 - 4y^2 - 16y - 5760 = 0
```

### Solving the Quartic Equation

We now have a quartic equation in terms of `y`

. Unfortunately, there's no general formula to solve quartic equations directly. However, we can try to factor it or use numerical methods to find the solutions.

**Step 4: Factorization (Optional)**

In this case, we can factor the equation:

```
(y - 12)(y + 12)(y^2 + 4y + 40) = 0
```

This gives us three possible solutions:

**y = 12****y = -12****y^2 + 4y + 40 = 0**

The quadratic equation `y^2 + 4y + 40 = 0`

has no real solutions.

**Step 5: Finding the values of x**

Since we defined `y = x + 3`

, we can find the values of `x`

:

**y = 12**: x = 12 - 3 =**9****y = -12**: x = -12 - 3 =**-15**

Therefore, the solutions to the equation (x+1)(x+3)(x+5)(x+7)=5760 are:

**x = 9** and **x = -15**.

### Important Note

The factorization in step 4 might not always be easy to find. For more complex quartic equations, you might need to use numerical methods or software tools to approximate the solutions.