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Let's work through each of the polynomial long division problems one by one, providing the quotient and the remainder for each. I'll start by solving the first few questions and then proceed through the rest. 

### 1. \( \frac{x^3 + 5x^2 + 7x + 2}{x + 2} \)
Performing polynomial long division:
1. Divide the leading term: \(\frac{x^3}{x} = x^2\).
2. Multiply and subtract: 
   \[
   (x^3 + 5x^2 + 7x + 2) - (x^2(x + 2)) = -x^2 + 7x + 2.
   \]
3. Repeat with the new polynomial:
   - Divide: \(\frac{-x^2}{x} = -x\).
   - Multiply and subtract:
   \[
   (-x^2 + 7x + 2) - (-x(x + 2)) = 9x + 2.
   \]
4. Divide again: \(\frac{9x}{x} = 9\).
5. Multiply and subtract:
   \[
   (9x + 2) - (9(x + 2)) = -16.
   \]
   
   The quotient is \( x^2 - x + 9 \) with a remainder of \(-16\), so the final answer is:
   \[
   \frac{x^3 + 5x^2 + 7x + 2}{x + 2} = x^2 - x + 9 - \frac{16}{x + 2}.
   \]

---

### 2. \( \frac{x^3 - 2x^2 - 5x + 6}{x - 3} \)
1. Divide: \(\frac{x^3}{x} = x^2\).
2. Multiply and subtract:
   \[
   (x^3 - 2x^2 - 5x + 6) - (x^2(x - 3)) = x^2 - 5x + 6.
   \]
3. Divide: \(\frac{x^2}{x} = x\).
4. Multiply and subtract:
   \[
   (x^2 - 5x + 6) - (x(x - 3)) = -2x + 6.
   \]
5. Divide: \(\frac{-2x}{x} = -2\).
6. Multiply and subtract:
   \[
   (-2x + 6) - (-2(x - 3)) = 0.
   \]
   
   The quotient is \( x^2 + x - 2 \) with no remainder, so the final answer is:
   \[
   \frac{x^3 - 2x^2 - 5x + 6}{x - 3} = x^2 + x - 2.
   \]

---

### 3. \( \frac{6x^3 + 7x^2 + 12x - 5}{3x - 1} \)
1. Divide: \(\frac{6x^3}{3x} = 2x^2\).
2. Multiply and subtract:
   \[
   (6x^3 + 7x^2 + 12x - 5) - (2x^2(3x - 1)) = 9x^2 + 12x - 5.
   \]
3. Divide: \(\frac{9x^2}{3x} = 3x\).
4. Multiply and subtract:
   \[
   (9x^2 + 12x - 5) - (3x(3x - 1)) = 15x - 5.
   \]
5. Divide: \(\frac{15x}{3x} = 5\).
6. Multiply and subtract:
   \[
   (15x - 5) - (5(3x - 1)) = 0.
   \]
   
   The quotient is \( 2x^2 + 3x + 5 \) with no remainder, so the final answer is:
   \[
   \frac{6x^3 + 7x^2 + 12x - 5}{3x - 1} = 2x^2 + 3x + 5.
   \]

---

### 4. \( \frac{6x^3 + 17x^2 + 27x + 20}{3x + 4} \)
1. Divide: \(\frac{6x^3}{3x} = 2x^2\).
2. Multiply and subtract:
   \[
   (6x^3 + 17x^2 + 27x + 20) - (2x^2(3x + 4)) = 9x^2 + 27x + 20.
   \]
3. Divide: \(\frac{9x^2}{3x} = 3x\).
4. Multiply and subtract:
   \[
   (9x^2 + 27x + 20) - (3x(3x + 4)) = 8.
   \]

   The quotient is \( 2x^2 + 3x \) with a remainder of \( 8 \), so the final answer is:
   \[
   \frac{6x^3 + 17x^2 + 27x + 20}{3x + 4} = 2x^2 + 3x + \frac{8}{3x + 4}.
   \]

---

### 5. \( \frac{12x^2 + x - 4}{3x - 2} \)
1. Divide: \(\frac{12x^2}{3x} = 4x\).
2. Multiply and subtract:
   \[
   (12x^2 + x - 4) - (4x(3x - 2)) = 9x - 4.
   \]
3. Divide: \(\frac{9x}{3x} = 3\).
4. Multiply and subtract:
   \[
   (9x - 4) - (3(3x - 2)) = 2.
   \]
   
   The quotient is \( 4x + 3 \) with a remainder of \( 2 \), so the final answer is:
   \[
   \frac{12x^2 + x - 4}{3x - 2} = 4x + 3 + \frac{2}{3x - 2}.
   \]

---

Would you like me to continue with the remaining questions or need detailed steps for any of these? Here are some follow-up questions related to these solutions:

1. How do you perform polynomial long division step-by-step?
2. What is the purpose of expressing the remainder as a fraction?
3. How do you check if your polynomial division result is correct?
4. What are other methods of dividing polynomials, such as synthetic division?
5. How does polynomial division relate to the factor theorem?

**Tip**: When dividing polynomials, always align terms by descending powers for clarity in the subtraction step.

Divide the polynomials using Long Division. If there is a remainder, express that remainder as a fraction added to the quotient. Problems: 1. x^3 + 5x^2 + 7x + 2 by x + 2, 2. x^3 - 2x^2 - 5x + 6 by x - 3, and so on.

Learn how to divide cubic polynomials by linear divisors using long division. The problem set covers polynomial division with remainders expressed as fractions. Suitable for students in Grades 10-12.

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