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To solve the expression \( a^2 + 2a + ab + b + 1 \), we need to factorize it. Let's proceed step by step:

The given expression is:

\[
a^2 + 2a + ab + b + 1
\]

### Step 1: Group terms
We can try to group the terms to factor more easily:

\[
a^2 + ab + 2a + b + 1
\]

### Step 2: Factor common terms
From the first two terms \( a^2 + ab \), we can factor out \( a \):

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

### Step 3: Rearrange and factor
Now we focus on factoring the remaining terms \( 2a + b + 1 \). While this does not immediately suggest a common factor, let's check the structure. Since the original question only asks to indicate one factor, the factor we've already found is \( (a + b) \).

Thus, **one factor** is:

\[
(a + b)
\]

Would you like me to continue exploring further factoring or details?

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Here are some related questions you might want to explore:

1. How do you fully factorize the expression \( a^2 + 2a + ab + b + 1 \)?
2. What are the steps to factor quadratic expressions in general?
3. Can you apply factorization by grouping for similar expressions?
4. How would you factorize an expression like \( a^2 + ab + 2a + b \)?
5. What are other methods to simplify algebraic expressions?

**Tip:** Always look for common factors in each term as a first step when attempting to factor algebraic expressions.

Factoriza: a^2 + 2a + ab + b + 1. Indica un factor.

Learn how to factor the algebraic expression a^2 + 2a + ab + b + 1 using factoring by grouping. This problem involves finding a common factor from grouped terms and is suitable for students in Grades 10-12.

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