[
= \left[(k+1)! - 1\right] + (k+1) \times (k+1)!
]

Factorisons ( (k+1)! ) :

[
= (k+1)! \times [1 + (k+1)] - 1
]

Let's factorize the expression \(\left[(k+1)! - 1\right] + (k+1) \times (k+1)!\) by focusing on the common factor \((k+1)!\):

Given expression:
\[
= \left[(k+1)! - 1\right] + (k+1) \times (k+1)!
\]

### Step 1: Distribute the terms
Notice that the expression can be separated into two parts:

\[
= \left[(k+1)!\right] + \left[(k+1) \times (k+1)!\right] - 1
\]

### Step 2: Factor out \((k+1)!\)
Now, factor out the common factor \((k+1)!\) from the first two terms:

\[
= (k+1)! \times \left[1 + (k+1)\right] - 1
\]

### Step 3: Simplify the expression inside the brackets
Simplify the expression inside the brackets:

\[
= (k+1)! \times \left[k+2\right] - 1
\]

So, the fully factorized expression is:

\[
= (k+1)! \times (k+2) - 1
\]

Would you like more details on this process or have any questions?

### Related Questions:
1. What is the factorial of a number, and how is it calculated?
2. How do you factorize expressions involving factorials?
3. What is the significance of the factorial function in combinatorics?
4. How can we simplify expressions that involve factorials and arithmetic operations?
5. What are some common mistakes to avoid when working with factorials in algebra?

### Tip:
When factorizing expressions, always look for common terms or factors that can be factored out to simplify the expression.

Explore the process of factorizing complex expressions involving factorials with a detailed step-by-step guide. This problem focuses on algebraic manipulation and the use of common factors, suitable for students in Grades 10-12.

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