To expand \( \frac{1}{(1 - x)^{3n + 3}} \), we can apply the **binomial series expansion** for expressions of the form \( (1 - x)^{-k} \), where \( k \) is a positive integer or a real number. 

The binomial series expansion for \( (1 - x)^{-k} \) is given by:

\[
(1 - x)^{-k} = \sum_{m=0}^{\infty} \binom{k+m-1}{m} x^m
\]

In this case, we have \( k = 3n + 3 \), so the expansion becomes:

\[
\frac{1}{(1 - x)^{3n+3}} = \sum_{m=0}^{\infty} \binom{3n+3+m-1}{m} x^m
\]

### Step-by-step expansion:

1. The general term of the expansion is:

   \[
   \text{Term} = \binom{3n+2+m}{m} x^m
   \]

   Here, \( \binom{3n+2+m}{m} \) is the binomial coefficient.

2. The first few terms of the expansion are:

   \[
   \frac{1}{(1 - x)^{3n+3}} = 1 + (3n+3) x + \frac{(3n+3)(3n+4)}{2!} x^2 + \frac{(3n+3)(3n+4)(3n+5)}{3!} x^3 + \cdots
   \]

### Conclusion:
This gives the infinite series representation of \( \frac{1}{(1 - x)^{3n+3}} \). Each term involves increasingly larger powers of \( x \) and binomial coefficients based on \( 3n + 3 \).

Would you like more details or clarifications?

### 5 Related Questions:
1. How is the binomial expansion applied to negative exponents?
2. Can you show the first four terms of the expansion for a specific value of \( n \)?
3. How does the convergence of this series depend on \( x \)?
4. What is the general term in a binomial expansion?
5. How can this series expansion be applied in practical problems?

### Tip:
For series expansions, always check the convergence radius to ensure the expansion is valid for the values of \( x \) you're working with.

Learn how to expand 1/(1 - x)^(3n+3) using the binomial series expansion. This problem covers the power series for negative exponents, providing step-by-step details of the expansion formula and key binomial coefficients. Suitable for advanced high school and college-level students.

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