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The problem in the image asks to divide \( 3x^4 - 5x^2 + 3 \) by \( x + 2 \). 

We can solve this using polynomial long division. Let's go step by step:

### Step 1: Set up the division

We are dividing:
\[
\frac{3x^4 - 5x^2 + 3}{x + 2}
\]

### Step 2: Perform polynomial long division

1. **Divide the first term** of the dividend \( 3x^4 \) by the first term of the divisor \( x \). This gives \( 3x^3 \).

2. **Multiply** \( 3x^3 \) by \( x + 2 \):
   \[
   3x^3 \cdot (x + 2) = 3x^4 + 6x^3
   \]

3. **Subtract** \( (3x^4 + 6x^3) \) from \( 3x^4 - 5x^2 + 3 \):
   \[
   (3x^4 - 5x^2 + 3) - (3x^4 + 6x^3) = -6x^3 - 5x^2 + 3
   \]

4. **Divide** the first term of the new dividend \( -6x^3 \) by \( x \), which gives \( -6x^2 \).

5. **Multiply** \( -6x^2 \) by \( x + 2 \):
   \[
   -6x^2 \cdot (x + 2) = -6x^3 - 12x^2
   \]

6. **Subtract** \( (-6x^3 - 12x^2) \) from \( -6x^3 - 5x^2 + 3 \):
   \[
   (-6x^3 - 5x^2 + 3) - (-6x^3 - 12x^2) = 7x^2 + 3
   \]

7. **Divide** the first term of the new dividend \( 7x^2 \) by \( x \), which gives \( 7x \).

8. **Multiply** \( 7x \) by \( x + 2 \):
   \[
   7x \cdot (x + 2) = 7x^2 + 14x
   \]

9. **Subtract** \( (7x^2 + 14x) \) from \( 7x^2 + 3 \):
   \[
   (7x^2 + 3) - (7x^2 + 14x) = -14x + 3
   \]

10. **Divide** the first term of the new dividend \( -14x \) by \( x \), which gives \( -14 \).

11. **Multiply** \( -14 \) by \( x + 2 \):
    \[
    -14 \cdot (x + 2) = -14x - 28
    \]

12. **Subtract** \( (-14x - 28) \) from \( -14x + 3 \):
    \[
    (-14x + 3) - (-14x - 28) = 31
    \]

### Step 3: Write the final result

The quotient is \( 3x^3 - 6x^2 + 7x - 14 \), and the remainder is 31. So, the result of the division is:

\[
\frac{3x^4 - 5x^2 + 3}{x + 2} = 3x^3 - 6x^2 + 7x - 14 + \frac{31}{x+2}
\]

Would you like more details or clarification on any part of this solution?

Here are 5 related questions:
1. How would the division change if the divisor were \( x - 2 \) instead of \( x + 2 \)?
2. Can synthetic division be used for this problem? Why or why not?
3. How do you handle division when there is a missing term in the polynomial?
4. What is the significance of the remainder in polynomial division?
5. How does the degree of the divisor and dividend affect the result?

**Tip:** In polynomial long division, always align like terms (same powers of \(x\)) to avoid mistakes when subtracting.

Learn how to perform polynomial long division of 3x^4 - 5x^2 + 3 by x + 2 in this detailed step-by-step guide. Suitable for high school algebra students.

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