To factor the expression \( 3a^2 + 3a + 2ab + 2b + ac + c \), we can group the terms in a way that makes factoring easier:

### Step 1: Group the terms
\[
(3a^2 + 3a) + (2ab + 2b) + (ac + c)
\]

### Step 2: Factor out the common factors from each group
1. From the first group \(3a^2 + 3a\), factor out \(3a\):
   \[
   3a(a + 1)
   \]

2. From the second group \(2ab + 2b\), factor out \(2b\):
   \[
   2b(a + 1)
   \]

3. From the third group \(ac + c\), factor out \(c\):
   \[
   c(a + 1)
   \]

### Step 3: Factor out the common binomial factor \((a + 1)\) from the entire expression:
\[
3a(a + 1) + 2b(a + 1) + c(a + 1)
\]

\[
(a + 1)(3a + 2b + c)
\]

### Final Answer:
The factored form of the expression \(3a^2 + 3a + 2ab + 2b + ac + c\) is:
\[
\boxed{(a + 1)(3a + 2b + c)}
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you factor a quadratic expression?
2. Can all polynomials be factored?
3. What are common factorization techniques?
4. How do you apply the distributive property in reverse for factoring?
5. How do you identify common factors in polynomial expressions?

**Tip:** When factoring, always look for common factors first to simplify the expression before attempting more complex methods.

Learn how to factor the expression 3a^2 + 3a + 2ab + 2b + ac + c with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring polynomials and binomial expansion. Suitable for students in Grades 9-12.

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