To solve this equation, we'll follow these steps:

1. Expand the left side of the equation:
   (x+1)(x+2)(x+3)(x+4) = (x^2 + 3x + 2)(x^2 + 5x + 6)
   = (x^2 + 3x + 2)(x^2 + 5x + 6) = x^4 + 5x^3 + 6x^2 + 3x^3 + 15x^2 + 18x + 2x^2 + 10x + 12
   = x^4 + 8x^3 + 23x^2 + 28x + 12

2. Rewrite the equation with the expanded left side:
   x^4 + 8x^3 + 23x^2 + 28x + 12 = x^6

3. Subtract x^4 from both sides to simplify the equation:
   8x^3 + 23x^2 + 28x + 12 = x^6 - x^4

4. Factor the right side of the equation using the difference of squares formula:
   x^4 - (x^2)^2 = (x^2 - x)(x^2 + x)
   So, the equation becomes:
   8x^3 + 23x^2 + 28x + 12 = (x^2 - x)(x^2 + x)

5. Multiply the right side of the equation:
   (x^2 - x)(x^2 + x) = x^4 - x^3 + x^3 - x^2
   = x^4 - x^2
6. Rewrite the equation with the simplified right side:
   8x^3 + 23x^2 + 28x + 12 = x^4 - x^2

7. Subtract x^4 from both sides to move all terms to the left side:
   8x^3 + 23x^2 + 28x + 12 - x^4 = - x^2

8. Combine like terms:
   -x^4 + 8x^3 + 23x^2 + 28x + 12 = 0
9. Factor the left side of the equation by grouping:
   -x^4 + 8x^3 + 23x^2 + 28x + 12 = -(x^4 - 8x^3) + (23x^2 + 28x + 12)
   = -x^3(x - 8) + (23x^2 + 28x + 12)

10. Factor the quadratic expression inside the parentheses:
   23x^2 + 28x + 12 = (x + 2)(23x + 6)
   So, the equation becomes:
   -x^3(x - 8) + (x + 2)(23x + 6) = 0
11. Distribute the negative sign to the terms inside the first set of parentheses:
   -x^3(x - 8) = -x^4 + 8x^3
   So, the equation becomes:
   -x^4 + 8x^3 + (x + 2)(23x + 6) = 0

12. Multiply the terms inside the parentheses:
   (x + 2)(23x + 6) = 23x^2 + 6x + 46x + 12
   = 23x^2 + 52x + 12

13. Rewrite the equation with the multiplied terms:
   -x^4 + 8x^3 + 23x^2 + 52x + 12 = 0
14. Combine like terms:
   -x^4 + 8x^3 + 23x^2 + 52x + 12 = -x^4 + 8x^3 + 23x^2 + 52x + 12
   = -(x^4 - 8x^3 - 23x^2 - 52x - 12)

15. Factor the left side of the equation by grouping:
   -(x^4 - 8x^3 - 23x^2 - 52x - 12) = -(x^4 - 8x^3) - (23x^2 + 52x + 12)
   = -x^3(x - 8) - (23x^2 + 52x + 12)

16. Factor the quadratic expression inside the parentheses:
   23x^2 + 52x + 12 = (23x + 3)(x + 4)
   So, the equation becomes:
   -x^3(x - 8) - (23x + 3)(x + 4) = 0
17. Distribute the negative sign to the terms inside the second set of parentheses:
   -(23x + 3)(x + 4) = -(23x^2 + 92x + 12)
   So, the equation becomes:
   -x^3(x - 8) - 23x^2 - 92x - 12 = 0

18. Rewrite the equation with the distributed terms:
   -x^4 + 8x^3 - 23x^2 - 92x - 12 = 0

19. Move all terms to the left side of the equation:
   -x^4 + 8x^3 - 23x^2 - 92x - 12 = 0

20. Add 23x^2 to both sides of the equation:
   -x^4 + 8x^3 - 23x^2 + 23x^2 - 92x - 12 = 0

21. Simplify the equation:
   -x^4 + 8x^3 - 92x - 12 = 0
22. Add 92x to both sides of the equation:
   -x^4 + 8x^3 + 92x - 92x - 12 = 0

23. Simplify the equation:
   -x^4 + 8x^3 + 92x - 92x - 12 = 0

24. Remove the 92x terms:
   -x^4 + 8x^3 - 12 = 0

25. Add 12 to both sides of the equation:
   -x^4 + 8x^3 + 12 - 12 = 0

26. Simplify the equation:
   -x^4 + 8x^3 = 0
27. Factor out the common term -x^3:
   -x^3(x) + 8x^3 = 0

28. Combine like terms:
   -x^3(x - 8) = 0

29. Set each factor equal to zero:
   -x^3 = 0 or x - 8 = 0

30. Solve for x:
   -x^3 = 0 -> x^3 = 0 -> x = 0
   x - 8 = 0 -> x = 8

Where is the first mistake