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To solve this polynomial division problem, we need to perform long division with the following expression:

\[
\frac{3x^4 - 7x^2 + 6x - 7}{x + 2}
\]

### Step-by-Step Long Division:

1. **Divide the first term:**  
   Divide the leading term of the numerator \(3x^4\) by the leading term of the denominator \(x\), giving \(3x^3\).
   
2. **Multiply and subtract:**  
   Multiply \(3x^3\) by \(x + 2\) to get \(3x^4 + 6x^3\).  
   Subtract this from the original polynomial:
   \[
   (3x^4 - 7x^2 + 6x - 7) - (3x^4 + 6x^3) = -6x^3 - 7x^2 + 6x - 7
   \]

3. **Repeat with the new leading term:**  
   Divide \(-6x^3\) by \(x\) to get \(-6x^2\).  
   Multiply \(-6x^2\) by \(x + 2\) to get \(-6x^3 - 12x^2\).  
   Subtract this from the current polynomial:
   \[
   (-6x^3 - 7x^2 + 6x - 7) - (-6x^3 - 12x^2) = 5x^2 + 6x - 7
   \]

4. **Continue to the next term:**  
   Divide \(5x^2\) by \(x\) to get \(5x\).  
   Multiply \(5x\) by \(x + 2\) to get \(5x^2 + 10x\).  
   Subtract this from the current polynomial:
   \[
   (5x^2 + 6x - 7) - (5x^2 + 10x) = -4x - 7
   \]

5. **Final step with the remaining terms:**  
   Divide \(-4x\) by \(x\) to get \(-4\).  
   Multiply \(-4\) by \(x + 2\) to get \(-4x - 8\).  
   Subtract this from the remaining polynomial:
   \[
   (-4x - 7) - (-4x - 8) = 1
   \]

### Result:
The quotient is \(3x^3 - 6x^2 + 5x - 4\) and the remainder is \(1\).

So,
\[
\text{Quotient} = 3x^3 - 6x^2 + 5x - 4, \quad \text{Remainder} = 1
\]

Would you like further explanation or have questions?

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**Related Practice Questions:**
1. How would you perform polynomial division for \((4x^4 + 8x^3 - 5x + 3) \div (x + 1)\)?
2. What is the result of dividing \((2x^5 - 4x^3 + 6x - 8) \div (x - 2)\)?
3. Can you find the remainder when dividing \((x^3 - 2x + 4) \div (x + 3)\)?
4. How does synthetic division compare to long division in terms of speed?
5. If the remainder is zero, what does this say about the divisor and the original polynomial?

**Tip:** Always align terms by descending powers to avoid errors in polynomial division.

Use long division to find the quotient and the remainder for the expression (3x^4 - 7x^2 + 6x - 7) / (x + 2).

Learn how to find the quotient and remainder for the polynomial division of (3x^4 - 7x^2 + 6x - 7) by (x + 2) using long division. Detailed steps are provided, covering key algebra concepts and polynomial division techniques. Ideal for high school students in Grades 10-12.

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