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The task is to divide two polynomials \( P(x) \) and \( D(x) \), which are given as:

\[
P(x) = 6x^4 - 3x^3 + 29x^2
\]
\[
D(x) = 3x^2 + 13
\]

We will use polynomial long division to find \( \frac{P(x)}{D(x)} \).

### Steps:

1. **Divide the leading term of \( P(x) \) by the leading term of \( D(x) \):**
   \[
   \frac{6x^4}{3x^2} = 2x^2
   \]

2. **Multiply \( 2x^2 \) by \( D(x) = 3x^2 + 13 \):**
   \[
   2x^2 \times (3x^2 + 13) = 6x^4 + 26x^2
   \]

3. **Subtract \( 6x^4 + 26x^2 \) from \( P(x) = 6x^4 - 3x^3 + 29x^2 \):**
   \[
   (6x^4 - 3x^3 + 29x^2) - (6x^4 + 26x^2) = -3x^3 + 3x^2
   \]

4. **Divide the leading term of the new polynomial \( -3x^3 \) by the leading term of \( D(x) \):**
   \[
   \frac{-3x^3}{3x^2} = -x
   \]

5. **Multiply \( -x \) by \( D(x) = 3x^2 + 13 \):**
   \[
   -x \times (3x^2 + 13) = -3x^3 - 13x
   \]

6. **Subtract \( -3x^3 - 13x \) from \( -3x^3 + 3x^2 \):**
   \[
   (-3x^3 + 3x^2) - (-3x^3 - 13x) = 3x^2 + 13x
   \]

7. **Divide the leading term \( 3x^2 \) by the leading term of \( D(x) \):**
   \[
   \frac{3x^2}{3x^2} = 1
   \]

8. **Multiply \( 1 \) by \( D(x) = 3x^2 + 13 \):**
   \[
   1 \times (3x^2 + 13) = 3x^2 + 13
   \]

9. **Subtract \( 3x^2 + 13 \) from \( 3x^2 + 13x \):**
   \[
   (3x^2 + 13x) - (3x^2 + 13) = 13x - 13
   \]

The remainder is \( 13x - 13 \).

### Final Result:
\[
\frac{P(x)}{D(x)} = 2x^2 - x + 1 + \frac{13x - 13}{D(x)}
\]
where \( D(x) = 3x^2 + 13 \).

Would you like further details or have any questions?

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Here are some follow-up questions for deeper understanding:

1. How does synthetic division differ from polynomial long division?
2. What are the applications of polynomial division in real-life problems?
3. Why is it necessary to subtract during the division process?
4. Can you perform polynomial division if the divisor is a higher degree than the dividend?
5. What are the different ways to express the remainder in polynomial division?

**Tip:** Polynomial division is essential for simplifying complex rational expressions and solving higher-degree equations.

Two polynomials P and D are given. Use either synthetic or long division to find P(x)/D(x). Given: P(x) = 6x^4 - 3x^3 + 29x^2, D(x) = 3x^2 + 13.

Learn how to perform polynomial long division to divide P(x) = 6x^4 - 3x^3 + 29x^2 by D(x) = 3x^2 + 13. This step-by-step solution covers polynomial division concepts, suitable for Grade 11-12 and college-level students.

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