https://file.aiquickdraw.com/mathbot/file/1729187029830jisrju4v.jpg

The problem asks to divide the polynomial \( 6x^3 - 7x^2 + 2 \) by \( 3x + 1 \).

This is a division of polynomials, and we can solve this using polynomial long division. Let's perform the steps of division.

### Step 1: Divide the leading term of the numerator by the leading term of the denominator.
The leading term in the numerator is \( 6x^3 \), and the leading term in the denominator is \( 3x \).

\[
\frac{6x^3}{3x} = 2x^2
\]

### Step 2: Multiply the result by the entire divisor \( 3x + 1 \).

\[
2x^2 \times (3x + 1) = 6x^3 + 2x^2
\]

### Step 3: Subtract this from the original polynomial.

\[
(6x^3 - 7x^2 + 2) - (6x^3 + 2x^2) = -9x^2 + 2
\]

### Step 4: Divide the new leading term \( -9x^2 \) by \( 3x \).

\[
\frac{-9x^2}{3x} = -3x
\]

### Step 5: Multiply the result by \( 3x + 1 \).

\[
-3x \times (3x + 1) = -9x^2 - 3x
\]

### Step 6: Subtract this from the new polynomial.

\[
(-9x^2 + 2) - (-9x^2 - 3x) = 3x + 2
\]

### Step 7: Divide \( 3x \) by \( 3x \).

\[
\frac{3x}{3x} = 1
\]

### Step 8: Multiply the result by \( 3x + 1 \).

\[
1 \times (3x + 1) = 3x + 1
\]

### Step 9: Subtract this from the current polynomial.

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

### Final Result:

The quotient is \( 2x^2 - 3x + 1 \) and the remainder is \( 1 \). Thus, the division is:

\[
\frac{6x^3 - 7x^2 + 2}{3x + 1} = 2x^2 - 3x + 1 + \frac{1}{3x + 1}
\]

This corresponds to the first option in the image:
\[
2x^2 - 3x + 1 + \frac{1}{3x + 1}
\]

Would you like more details on any of these steps?

Here are 5 related questions:
1. How does synthetic division differ from long division for polynomials?
2. Can polynomial division ever result in a remainder of 0?
3. How does the degree of the divisor affect the division process?
4. What is the connection between polynomial division and factorization?
5. How can you check if a division result is correct?

**Tip:** To verify polynomial division results, you can multiply the quotient by the divisor and add the remainder to see if it equals the original polynomial.

Divide the following polynomials: (6x^3 - 7x^2 + 2) / (3x + 1)

Learn how to divide the polynomial 6x^3 - 7x^2 + 2 by 3x + 1 using polynomial long division. This detailed solution demonstrates the process, including key algebraic concepts and polynomial division techniques. Suitable for Grades 9-11 students.

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