Prove that `t^(2) + 3t + 3` is a factor of `( t+1)^(n+1) + (t+2)^(2n -1)` for all intergral values of n `in` N.

Explanation

To prove that \( t^2 + 3t + 3 \) is a factor of \( (t+1)^{n+1} + (t+2)^{2n-1} \) for all integral values of \( n \in \mathbb{N} \), we can follow these steps:

### Step 1: Rewrite the Polynomial
We start by rewriting \( t^2 + 3t + 3 \) in a more manageable form. We can express it as:
\[ t^2 + 3t + 3 = (t+1)^2 + (t+1) + 1 \]
Let \( z = t + 1 \). Then, we have:
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