https://file.aiquickdraw.com/mathbot/file/1727472572699ve623v0c.jpg

The division problem shown in the image is:

\[
\frac{12x^2 + 29x + 15}{3x + 2}
\]

We need to perform polynomial division, where \( 12x^2 + 29x + 15 \) is the dividend and \( 3x + 2 \) is the divisor. Let's proceed with the division to find the quotient and remainder.

### Step-by-step division:

1. **Divide the first term of the dividend by the first term of the divisor**:
   \[
   \frac{12x^2}{3x} = 4x
   \]
   So, the first term of the quotient is \( 4x \).

2. **Multiply \( 4x \) by \( 3x + 2 \)**:
   \[
   4x(3x + 2) = 12x^2 + 8x
   \]

3. **Subtract** \( 12x^2 + 8x \) from the dividend:
   \[
   (12x^2 + 29x + 15) - (12x^2 + 8x) = 21x + 15
   \]

4. **Divide the first term of the new polynomial by the first term of the divisor**:
   \[
   \frac{21x}{3x} = 7
   \]
   So, the next term in the quotient is \( 7 \).

5. **Multiply \( 7 \) by \( 3x + 2 \)**:
   \[
   7(3x + 2) = 21x + 14
   \]

6. **Subtract** \( 21x + 14 \) from \( 21x + 15 \):
   \[
   (21x + 15) - (21x + 14) = 1
   \]

Thus, the quotient is \( 4x + 7 \) and the remainder is \( 1 \).

### Final Answer:
- **Quotient**: \( 4x + 7 \)
- **Remainder**: \( 1 \)

Would you like further details on any step or have additional questions?

Here are some related questions:
1. How can we verify the result of a polynomial division?
2. What are the conditions for a remainder of zero in polynomial division?
3. Can synthetic division be used for this type of division?
4. How does long division of polynomials relate to factorization?
5. What if the divisor had been a quadratic instead of a linear expression?

**Tip**: When dividing polynomials, ensure you carefully align like terms and perform accurate subtraction at each step to avoid errors.

Divide: (12x^2 + 29x + 15) ÷ (3x + 2). Your answer should give the quotient and the remainder.

Learn how to divide polynomials: (12x^2 + 29x + 15) ÷ (3x + 2). This problem involves polynomial long division and provides a step-by-step solution to find the quotient and remainder. Suitable for students in Grades 9-12.

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