We will solve the division of the polynomials:

\[
\frac{5x^4 - 3x^3 + 2x^2 - 5}{x^2 + 4}
\]

### Steps:
1. **Set up the long division of polynomials**:
   \[
   \text{Divide the highest degree term first:} \frac{5x^4}{x^2} = 5x^2
   \]

2. **Multiply** \( 5x^2 \) by the divisor \( x^2 + 4 \):
   \[
   5x^2 \times (x^2 + 4) = 5x^4 + 20x^2
   \]

3. **Subtract**:
   \[
   (5x^4 - 3x^3 + 2x^2 - 5) - (5x^4 + 20x^2) = -3x^3 - 18x^2 - 5
   \]

4. **Divide** \( -3x^3 \) by \( x^2 \):
   \[
   \frac{-3x^3}{x^2} = -3x
   \]

5. **Multiply** \( -3x \) by the divisor \( x^2 + 4 \):
   \[
   -3x \times (x^2 + 4) = -3x^3 - 12x
   \]

6. **Subtract**:
   \[
   (-3x^3 - 18x^2 - 5) - (-3x^3 - 12x) = -18x^2 + 12x - 5
   \]

7. **Divide** \( -18x^2 \) by \( x^2 \):
   \[
   \frac{-18x^2}{x^2} = -18
   \]

8. **Multiply** \( -18 \) by the divisor \( x^2 + 4 \):
   \[
   -18 \times (x^2 + 4) = -18x^2 - 72
   \]

9. **Subtract**:
   \[
   (-18x^2 + 12x - 5) - (-18x^2 - 72) = 12x + 67
   \]

### Final result:

\[
\frac{5x^4 - 3x^3 + 2x^2 - 5}{x^2 + 4} = 5x^2 - 3x - 18 + \frac{12x + 67}{x^2 + 4}
\]

This is the quotient: \( 5x^2 - 3x - 18 \), and the remainder is \( \frac{12x + 67}{x^2 + 4} \).

Would you like further clarification or details?

### Related Questions:
1. How do you perform synthetic division on polynomials?
2. Can the remainder be simplified further in polynomial division?
3. What is the significance of the remainder in polynomial long division?
4. How can you factor the quotient of a division?
5. What are other methods to divide polynomials?

**Tip**: When dividing polynomials, always organize terms by descending powers for easier manipulation.

(5x^4 − 3x^3 + 2x^2 − 5) ÷ (x^2 + 4)

Learn how to solve the polynomial division problem (5x^4 − 3x^3 + 2x^2 − 5) ÷ (x^2 + 4) with a detailed, step-by-step breakdown. Understand key concepts like polynomial division and theorems behind long division of polynomials. Suitable for high school students in Grades 10-12.

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