## Solving the Equation (x+1)(x+2)(x+3)(x+4) = 120

This equation involves a product of four consecutive terms equaling a constant. To solve it, we can employ a combination of algebraic manipulation and strategic thinking.

**1. Recognizing the Pattern:**

The left-hand side of the equation represents the product of four consecutive integers. We can use this pattern to our advantage.

**2. Factoring 120:**

Start by factoring 120 into its prime factors: 120 = 2 x 2 x 2 x 3 x 5

**3. Finding Consecutive Integers:**

Now, try to group these prime factors in a way that forms four consecutive integers. Observe that:

- 2 x 3 = 6
- 2 x 5 = 10

These are two consecutive integers. The other two consecutive integers can be formed by subtracting 1 and adding 1:

- 6 - 1 = 5
- 10 + 1 = 11

Therefore, we have found the four consecutive integers: **5, 6, 10, and 11.**

**4. Verification:**

Let's verify our solution:

(5 + 1)(5 + 2)(5 + 3)(5 + 4) = 6 x 7 x 8 x 9 = 3024

This doesn't match the original equation. This indicates that our initial assumption that the product of four consecutive integers must be equal to 120 is incorrect.

**5. Finding the Solution:**

Since the initial assumption was incorrect, we need to solve the equation directly.

**Here's how we can solve the equation (x+1)(x+2)(x+3)(x+4) = 120:**

**Expand the product:**Expand the left side of the equation. This will result in a fourth-degree polynomial.**Rearrange:**Move all terms to one side to get a polynomial equation equal to zero.**Factoring:**Attempt to factor the polynomial. This might be challenging, but using techniques like the Rational Root Theorem can be helpful.**Solve for x:**Once you have factored the polynomial, set each factor equal to zero and solve for the individual values of x.

**Note:** The equation (x+1)(x+2)(x+3)(x+4) = 120 might not have rational solutions. You might need to use numerical methods or graphing techniques to find the approximate solutions.

**In conclusion:**

While the initial approach of finding consecutive integers might seem promising, it does not yield the correct solution. To solve the equation (x+1)(x+2)(x+3)(x+4) = 120, you will need to expand the equation, rearrange it, and attempt to factor the resulting polynomial.