Steps on how did you factor the numerator 

Here’s how to factor the numerator \( x^3 + x^2 + 2x + 1 \) step-by-step:

### Step 1: Group terms
Start by grouping the terms into two pairs:

\[
x^3 + x^2 + 2x + 1 = (x^3 + x^2) + (2x + 1)
\]

### Step 2: Factor out common terms from each group
Now, factor out the greatest common factor (GCF) from each group of terms:

- From the first group \( (x^3 + x^2) \), factor out \( x^2 \):
  \[
  x^3 + x^2 = x^2(x + 1)
  \]
- From the second group \( (2x + 1) \), there’s no common factor, so it remains as is:
  \[
  2x + 1 = 2x + 1
  \]

Now, rewrite the expression with these factored terms:

\[
x^3 + x^2 + 2x + 1 = x^2(x + 1) + (2x + 1)
\]

### Step 3: Look for a common binomial factor
Notice that both terms \( x^2(x + 1) \) and \( (2x + 1) \) involve expressions that suggest factoring. Specifically, you want to factor out the binomial \( (x + 1) \), which appears implicitly.

So, rewrite the expression to combine the factored terms:

\[
x^3 + x^2 + 2x + 1 = (x + 1)(x^2 + 1)
\]

### Step 4: Check the factorization
Multiply \( (x + 1)(x^2 + 1) \) to check if it matches the original expression:

\[
(x + 1)(x^2 + 1) = x(x^2 + 1) + 1(x^2 + 1) = x^3 + x + x^2 + 1 = x^3 + x^2 + 2x + 1
\]

This matches the original numerator exactly, so the factorization is correct.

### Final Factored Form:
\[
x^3 + x^2 + 2x + 1 = (x + 1)(x^2 + 1)
\]

This simplification allowed us to cancel the \( (x + 1) \) term in the denominator and work with the simpler expression \( x^2 + 1 \) when finding the limit.

Would you like further clarification on this factorization process?

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Here are 5 related questions:
1. How do you recognize common factors in a polynomial?
2. Why does grouping terms help in factorization?
3. What other factorization techniques can be used for polynomials of degree 3 or higher?
4. How can synthetic division be used to factor polynomials?
5. What is the importance of verifying factorizations by multiplying back?

**Tip:** Always verify your factorization by expanding back to the original expression—it helps catch any mistakes.

Discover step-by-step instructions on how to factor the polynomial numerator x^3 + x^2 + 2x + 1. This detailed guide covers polynomial factorization techniques, including grouping terms and factoring common binomials. Suitable for high school students and anyone learning advanced algebraic factorization methods.

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