To solve the expression \(\frac{2x^4 + 3x^3 - x^2 - 8x + 5}{x + 2}\), we will perform polynomial long division.

### Step 1: Set up the division
We are dividing the polynomial \(2x^4 + 3x^3 - x^2 - 8x + 5\) by \(x + 2\).

### Step 2: Start dividing
1. Divide the leading term \(2x^4\) by the leading term \(x\). This gives \(2x^3\).
2. Multiply \(2x^3\) by \(x + 2\), which gives:
   \[
   (2x^3)(x + 2) = 2x^4 + 4x^3
   \]
3. Subtract \(2x^4 + 4x^3\) from the original polynomial:
   \[
   (2x^4 + 3x^3 - x^2 - 8x + 5) - (2x^4 + 4x^3) = -x^3 - x^2 - 8x + 5
   \]

### Step 3: Continue dividing
1. Divide the new leading term \(-x^3\) by \(x\). This gives \(-x^2\).
2. Multiply \(-x^2\) by \(x + 2\), which gives:
   \[
   (-x^2)(x + 2) = -x^3 - 2x^2
   \]
3. Subtract \(-x^3 - 2x^2\) from the current polynomial:
   \[
   (-x^3 - x^2 - 8x + 5) - (-x^3 - 2x^2) = x^2 - 8x + 5
   \]

### Step 4: Continue dividing
1. Divide the new leading term \(x^2\) by \(x\). This gives \(x\).
2. Multiply \(x\) by \(x + 2\), which gives:
   \[
   (x)(x + 2) = x^2 + 2x
   \]
3. Subtract \(x^2 + 2x\) from the current polynomial:
   \[
   (x^2 - 8x + 5) - (x^2 + 2x) = -10x + 5
   \]

### Step 5: Continue dividing
1. Divide the new leading term \(-10x\) by \(x\). This gives \(-10\).
2. Multiply \(-10\) by \(x + 2\), which gives:
   \[
   (-10)(x + 2) = -10x - 20
   \]
3. Subtract \(-10x - 20\) from the current polynomial:
   \[
   (-10x + 5) - (-10x - 20) = 25
   \]

### Final result:
The quotient is \(2x^3 - x^2 + x - 10\) and the remainder is \(25\).

Thus, the result of the division is:
\[
\frac{2x^4 + 3x^3 - x^2 - 8x + 5}{x + 2} = 2x^3 - x^2 + x - 10 + \frac{25}{x + 2}
\]

Would you like more details or have any questions?

Here are some related questions:
1. How does synthetic division differ from long division in polynomials?
2. Can every polynomial be factored completely, or do some have remainders?
3. What are the practical uses of dividing polynomials in real-world problems?
4. How do we identify the remainder in polynomial division?
5. What methods exist to verify the correctness of a division?

**Tip:** Always check your remainder when dividing polynomials, as it helps verify the accuracy of your result!

Learn how to divide the polynomial 2x⁴ + 3x³ - x² - 8x + 5 by x + 2 using long division. This problem demonstrates key algebraic concepts such as polynomial division and the division algorithm, suitable for students in Grades 10-12.

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