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**Hint**: In order to find one root for the equation $ \left( {x + 1} \right)\left( {x + 3} \right)\left( {x + 2} \right)\left( {x + 4} \right) = 120 $ , check out the number of terms in the right-hand side and compare it with the other side. If the number of terms is not the same then split the splittable value to make the terms equal on both the sides, arrange them in ascending or descending order, then compare the equations, and get the results.

**:**

__Complete step-by-step answer__We are given the equation $ \left( {x + 1} \right)\left( {x + 3} \right)\left( {x + 2} \right)\left( {x + 4} \right) = 120 $ .

Arranging the terms on the left side in ascending order, and we get:

$ \left( {x + 1} \right)\left( {x + 2} \right)\left( {x + 3} \right)\left( {x + 4} \right) = 120 $

Since, there are four terms on the left side and one term on the right side, we will split the left term that is $ 120 $ into parts.

As we know that $ 120 $ is the factorial of $ 5 $ . So, it can be written as $ 5! = 120 $ , and $ 5! $ can be splitted as $ 120 = 5! = 5 \times 4 \times 3 \times 2 \times 1 $ .

Substituting this value in the equation $ \left( {x + 1} \right)\left( {x + 2} \right)\left( {x + 3} \right)\left( {x + 4} \right) = 120 $ , we get:

$ \left( {x + 1} \right)\left( {x + 2} \right)\left( {x + 3} \right)\left( {x + 4} \right) = 5 \times 4 \times 3 \times 2 $ (Excluding $ 1 $ as anything multiplied to $ 1 $ , gives the same value).

Since, the left side is in increasing order and the right side is in decreasing order, so writing the right side also in increasing order and, we can write it as:

$ \left( {x + 1} \right)\left( {x + 2} \right)\left( {x + 3} \right)\left( {x + 4} \right) = 2 \times 3 \times 4 \times 5 $

By comparing each term of both the sides we can get all the roots, but since we want only one root.

So, taking one value:

$ \left( {x + 1} \right) = 2 $

Subtracting both the sides by $ 1 $ :

$

x + 1 - 1 = 2 - 1 \\

\Rightarrow x = 1 \;

$

Which is the one root obtained.

Therefore, one root of the equation $ \left( {x + 1} \right)\left( {x + 3} \right)\left( {x + 2} \right)\left( {x + 4} \right) = 120 $ is $ 1 $ .

**So, the correct answer is “1”.**

**Note**: Factorial of a number is the product of all the numbers starting from $ 1 $ and the number taken.

It’s important to arrange the both side values in the same order to find the roots easily.

We can check our answer by substituting the result into the given equation.

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