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

This article will guide you through solving the equation (x+1)(2x+6)=2(x+3)(1+x). We will use algebraic manipulations to simplify the equation and find the solutions.

### Expanding the Equation

First, we need to expand both sides of the equation by multiplying out the brackets:

**Left side:**(x+1)(2x+6) = 2x² + 6x + 2x + 6 = 2x² + 8x + 6**Right side:**2(x+3)(1+x) = 2(x + 3x + 3 + 3x) = 2(4x + 3) = 8x + 6

Now the equation becomes: 2x² + 8x + 6 = 8x + 6

### Simplifying the Equation

Next, we can simplify the equation by subtracting 8x and 6 from both sides:

2x² + 8x + 6 - 8x - 6 = 8x + 6 - 8x - 6

This leaves us with: 2x² = 0

### Solving for x

Finally, we can solve for x by dividing both sides by 2:

2x²/2 = 0/2

This gives us: x² = 0

Taking the square root of both sides: √(x²) = √(0)

Therefore, the solution to the equation is: **x = 0**

### Conclusion

We have successfully solved the equation (x+1)(2x+6)=2(x+3)(1+x) and found that the only solution is **x = 0**. This demonstrates that despite the initial complexity of the equation, through algebraic manipulation and simplification, we can arrive at a clear solution.