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The given expression is:

\[
f(x) = \frac{2x^3 - 5x^2 - 8x + 6}{x + 2}
\]

This appears to be a polynomial division problem where you are required to divide the cubic polynomial \(2x^3 - 5x^2 - 8x + 6\) by \(x + 2\).

Let's go ahead and perform the division.

### Step 1: Divide the first term of the numerator by the first term of the denominator.
\[
\frac{2x^3}{x} = 2x^2
\]

### Step 2: Multiply \(2x^2\) by \(x + 2\):
\[
2x^2(x + 2) = 2x^3 + 4x^2
\]

### Step 3: Subtract the result from the original polynomial:
\[
(2x^3 - 5x^2 - 8x + 6) - (2x^3 + 4x^2) = -9x^2 - 8x + 6
\]

### Step 4: Divide the first term of the new polynomial by the first term of the denominator:
\[
\frac{-9x^2}{x} = -9x
\]

### Step 5: Multiply \(-9x\) by \(x + 2\):
\[
-9x(x + 2) = -9x^2 - 18x
\]

### Step 6: Subtract the result from the current polynomial:
\[
(-9x^2 - 8x + 6) - (-9x^2 - 18x) = 10x + 6
\]

### Step 7: Divide the first term of the new polynomial by the first term of the denominator:
\[
\frac{10x}{x} = 10
\]

### Step 8: Multiply \(10\) by \(x + 2\):
\[
10(x + 2) = 10x + 20
\]

### Step 9: Subtract the result from the current polynomial:
\[
(10x + 6) - (10x + 20) = -14
\]

### Final result:
The quotient is \(2x^2 - 9x + 10\) and the remainder is \(-14\). So, the division can be expressed as:
\[
f(x) = 2x^2 - 9x + 10 + \frac{-14}{x + 2}
\]

Let me know if you would like more details, or if you have any questions!

Here are 5 related questions to explore further:
1. How do you perform polynomial long division for higher-degree polynomials?
2. What is the significance of the remainder in polynomial division?
3. How do you check if a given binomial is a factor of a polynomial?
4. How do synthetic division and long division differ in polynomial division?
5. Can you apply this method to divide by higher-degree polynomials?

**Tip:** Always check your final result by multiplying the quotient with the divisor and adding the remainder; it should give you the original polynomial.

Divide the polynomial 2x^3 - 5x^2 - 8x + 6 by x + 2.

Learn how to perform polynomial division of 2x^3 - 5x^2 - 8x + 6 by x + 2 with a detailed step-by-step breakdown. This solution explains polynomial long division, providing insights into the quotient and remainder calculations.

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