`x^3-3x^2+3x+7`

Explanation

To factorize the polynomial \( x^3 - 3x^2 + 3x + 7 \), we can follow these steps:

### Step 1: Identify the structure
We notice that the polynomial resembles the form of a cubic expression. We can use the identity for the sum and difference of cubes.

### Step 2: Rewrite the polynomial
We can rewrite the polynomial as:
\[ x^3 - 3x^2 + 3x + 7 = x^3 - 3x^2 + 3x + 1 + 6 \]
This allows us to group the terms as:
\[ (x^3 - 3x^2 + 3x + 1) + 6 \]

### Step 3: Recognize the cubic form
The first part, \( x^3 - 3x^2 + 3x + 1 \), can be recognized as:
\[ x^3 - 3(1)x^2 + 3(1)x + 1 \]
This matches the form \( A^3 - 3A^2B + 3AB^2 + B^3 \) where \( A = x \) and \( B = 1 \).

### Step 4: Factor using the identity
Using the identity for the sum of cubes, we can factor:
\[ x^3 - 3x^2 + 3x + 1 = (x - 1)^3 \]
Thus, we have:
\[ (x - 1)^3 + 6 \]

### Step 5: Final expression
Now, we can write the entire polynomial as:
\[ (x - 1)^3 + 6 \]
This is the factored form of the polynomial.

### Step 6: Conclusion
The polynomial \( x^3 - 3x^2 + 3x + 7 \) can be expressed as:
\[ (x - 1)^3 + 6 \]