Solving the Equation (x+1)(x+2)(x+3)(x+4) = 840
This equation presents an interesting challenge due to its form. Let's explore how to solve it:
Understanding the Equation
The equation represents the product of four consecutive integers equaling 840. This hints at a potential solution involving factorization.
Factorization Approach

Prime Factorization of 840: Begin by finding the prime factorization of 840: 840 = 2 x 2 x 2 x 3 x 5 x 7.

Grouping for Consecutive Integers: Look for combinations of these prime factors that can be grouped to form four consecutive integers. Notice that 5 x 6 x 7 x 8 = 840.

Solution: Therefore, we have:
 x + 1 = 5
 x + 2 = 6
 x + 3 = 7
 x + 4 = 8
 Solving for x, we get x = 4.
Verification
Substituting x = 4 back into the original equation: (4 + 1)(4 + 2)(4 + 3)(4 + 4) = 5 x 6 x 7 x 8 = 840. This confirms our solution.
Conclusion
By utilizing prime factorization and careful observation, we successfully found the solution to the equation (x + 1)(x + 2)(x + 3)(x + 4) = 840. This approach demonstrates the importance of recognizing patterns and applying appropriate techniques to solve algebraic equations.