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The given problem is asking to factor the polynomial \( x^3 + x^2 + 9x + 9 \) completely. 

We can approach this by grouping the terms in pairs:

\[
x^3 + x^2 + 9x + 9
\]

### Step 1: Group the terms
Group the polynomial in pairs to factor common terms:

\[
(x^3 + x^2) + (9x + 9)
\]

### Step 2: Factor out the greatest common factor (GCF) from each group
For the first group, \(x^3 + x^2\), the GCF is \(x^2\). For the second group, \(9x + 9\), the GCF is \(9\):

\[
x^2(x + 1) + 9(x + 1)
\]

### Step 3: Factor out the common binomial factor
Both terms now contain the factor \((x + 1)\), so we can factor this out:

\[
(x^2 + 9)(x + 1)
\]

### Step 4: Check if any further factoring is possible
The expression \(x^2 + 9\) is a sum of squares, which cannot be factored further over the real numbers. Therefore, the completely factored form is:

\[
(x^2 + 9)(x + 1)
\]

### Final Answer:
\[
(x^2 + 9)(x + 1)
\]

Would you like any further explanation on this factorization? Here are some additional questions related to this topic:

1. How do you factor other cubic polynomials using grouping?
2. Why can't the expression \(x^2 + 9\) be factored further in real numbers?
3. What are some common techniques to factor higher-degree polynomials?
4. How can you verify that the factorization is correct?
5. What are imaginary roots, and how do they relate to sums of squares like \(x^2 + 9\)?

**Tip:** Always start by checking for common factors before applying more advanced factoring techniques!

Factor the polynomial x^3 + x^2 + 9x + 9 completely.

Learn how to factor the cubic polynomial x^3 + x^2 + 9x + 9 using the factoring by grouping method. This step-by-step guide covers algebraic concepts like grouping and factoring by the greatest common factor (GCF). Suitable for students in Grades 9-12.

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