The given expression is:

\[
\frac{6x^3 - 9x^2 + 0x - 8}{3x^2 + 2}
\]

### Step 1: Perform Polynomial Long Division
We divide \(6x^3 - 9x^2 + 0x - 8\) by \(3x^2 + 2\).

1. Divide the leading term \(6x^3\) by \(3x^2\), which gives \(2x\).
   - Multiply \(2x\) by \(3x^2 + 2\): 
     \[
     2x \cdot (3x^2 + 2) = 6x^3 + 4x
     \]
   - Subtract \((6x^3 + 4x)\) from \((6x^3 - 9x^2 + 0x - 8)\):
     \[
     (6x^3 - 9x^2 + 0x - 8) - (6x^3 + 4x) = -9x^2 - 4x - 8
     \]

2. Divide the leading term \(-9x^2\) by \(3x^2\), which gives \(-3\).
   - Multiply \(-3\) by \(3x^2 + 2\): 
     \[
     -3 \cdot (3x^2 + 2) = -9x^2 - 6
     \]
   - Subtract \((-9x^2 - 6)\) from \((-9x^2 - 4x - 8)\):
     \[
     (-9x^2 - 4x - 8) - (-9x^2 - 6) = -4x - 2
     \]

### Step 2: Write the result
The quotient is \(2x - 3\) and the remainder is \(-4x - 2\). Thus, the division result can be expressed as:

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

This is the simplified form of the given division.

Would you like further details or explanations?

Here are five related questions you might explore:
1. How do you handle division when the remainder is zero?
2. What is the significance of the remainder in polynomial division?
3. How do you divide polynomials using synthetic division?
4. Can you divide polynomials with more complicated denominators?
5. How do you apply polynomial division in solving algebraic equations?

**Tip:** Always verify the result of polynomial division by multiplying the quotient and divisor to see if you get back the original dividend!

Learn how to solve the polynomial long division of (6x^3 - 9x^2 - 8) ÷ (3x^2 + 2) with a step-by-step breakdown. This problem explains key algebraic concepts such as polynomial long division and understanding remainders. Suitable for students in Grades 10-12.

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