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Step 1:I'll solve this polynomial factorization problem step by step using the grouping method:
Step 2:: Rearrange the polynomial terms
First, let's write out the original polynomial: $$x^{3} + x^{2} + 3x + 3
Step 3:: Group the first two and last two terms
\left(x^{3} + x^{2}\right) + \left(3x + 3\right)
Step 4:: Factor out the common factor from the first group
x^{2}\left(x + 1\right) + 3\left(x + 1\right)
Step 5:: Identify the common factor
Notice that $\left(x + 1\right)$ is now a common factor in both groups.
Step 6:: Factor out the common factor
\left(x + 1\right)\left(x^{2} + 3\right)
Final Answer
Key Insights: - The grouping method works by finding a common factor across grouped terms - In this case, $\left(x + 1\right)$ was the key to successful factorization - The polynomial cannot be factored further over the real numbers
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