## Solving the Equation (x+1)(x+2)(2x+1)(2x+3)=6

This equation involves a product of four linear factors equaling a constant, leading to a quartic equation. Here's how we can solve it:

### 1. Expand and Simplify

First, we expand the product on the left-hand side:

**Step 1:**Expand the first two factors: (x+1)(x+2) = x² + 3x + 2**Step 2:**Expand the last two factors: (2x+1)(2x+3) = 4x² + 8x + 3**Step 3:**Multiply the results from Step 1 and Step 2: (x² + 3x + 2)(4x² + 8x + 3) = 4x⁴ + 20x³ + 35x² + 26x + 6

Now the equation becomes:

4x⁴ + 20x³ + 35x² + 26x + 6 = 6

### 2. Rearrange and Solve

Subtract 6 from both sides to get a standard quartic equation:

4x⁴ + 20x³ + 35x² + 26x = 0

Now we can factor out a common factor of 2x:

2x(2x³ + 10x² + 17.5x + 13) = 0

This gives us one solution immediately:

**x = 0**

To find the remaining solutions, we need to solve the cubic equation:

2x³ + 10x² + 17.5x + 13 = 0

### 3. Solving the Cubic Equation

Solving cubic equations can be challenging. Here are some approaches:

**Rational Root Theorem:**This theorem helps us find potential rational roots. However, it doesn't guarantee finding a solution in all cases.**Numerical Methods:**Methods like Newton-Raphson iteration can approximate the solutions numerically.**Factoring:**In some cases, the cubic equation can be factored, but this is not always possible.

For this specific equation, it's difficult to find exact solutions using elementary methods. We can resort to numerical methods or graphing tools to approximate the solutions.

### 4. Approximate Solutions

Using a numerical solver or graphing tools, we find that the cubic equation has three real roots:

**x ≈ -1.5****x ≈ -2.5****x ≈ -1**

Therefore, the equation (x+1)(x+2)(2x+1)(2x+3)=6 has the following solutions:

**x = 0****x ≈ -1.5****x ≈ -2.5****x ≈ -1**