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We are asked to find the quotient of the following polynomial division:

\[
\frac{x^3 + 6x^2 + 11x + 6}{x^2 + 4x + 3}
\]

Let's solve this using polynomial long division.

### Step 1: Divide the leading term
The first term of the dividend \(x^3 + 6x^2 + 11x + 6\) is \(x^3\) and the leading term of the divisor \(x^2 + 4x + 3\) is \(x^2\). Divide \(x^3\) by \(x^2\) to get the first term of the quotient:

\[
\frac{x^3}{x^2} = x
\]

### Step 2: Multiply and subtract
Now multiply \(x\) by the entire divisor \(x^2 + 4x + 3\):

\[
x(x^2 + 4x + 3) = x^3 + 4x^2 + 3x
\]

Now subtract this result from the original polynomial:

\[
(x^3 + 6x^2 + 11x + 6) - (x^3 + 4x^2 + 3x) = (6x^2 - 4x^2) + (11x - 3x) + 6 = 2x^2 + 8x + 6
\]

### Step 3: Repeat the process
Now divide the new leading term \(2x^2\) by \(x^2\) to get the next term of the quotient:

\[
\frac{2x^2}{x^2} = 2
\]

Multiply 2 by the divisor:

\[
2(x^2 + 4x + 3) = 2x^2 + 8x + 6
\]

Now subtract this from the current remainder:

\[
(2x^2 + 8x + 6) - (2x^2 + 8x + 6) = 0
\]

### Final Quotient
The remainder is zero, and the quotient is:

\[
x + 2
\]

Thus, the correct answer is:

\[
\boxed{x + 2}
\]

Would you like further details or explanations?

Here are five related questions to expand on this concept:
1. How does polynomial long division differ from numerical long division?
2. Can you always perform polynomial division when the degrees of the polynomials are the same?
3. What happens if the remainder is not zero in a polynomial division?
4. How does synthetic division simplify the division of polynomials?
5. What are the applications of polynomial division in real-world scenarios?

**Tip:** Always align terms according to their degree (power of \(x\)) when performing polynomial long division to avoid mistakes.

What is the quotient of (x^3 + 6x^2 + 11x + 6) ÷ (x^2 + 4x + 3)?

Learn how to solve polynomial division problems, such as finding the quotient of (x^3 + 6x^2 + 11x + 6) ÷ (x^2 + 4x + 3), using long division. This detailed explanation covers key algebraic concepts and formulas for students in Grades 9-12.

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