Let
\[a(x)=-7x^5+3x^3-6x-8\], and
\[b(x)=x^2\].
When dividing
\[a\] by
\[b\], we can find the unique quotient polynomial
\[q\] and remainder polynomial
\[r\] that satisfy the following equation:

\[\dfrac{a(x)}{b(x)}=q(x) + \dfrac{r(x)}{b(x)}\],
where the degree of
\[r(x)\] is less than the degree of
\[b(x)\].
What is the quotient,
\[q(x)\]?
\[ q(x)=\]
What is the remainder,
\[r(x)\]?
\[r(x)=\]

To solve the division of the polynomial \( a(x) = -7x^5 + 3x^3 - 6x - 8 \) by \( b(x) = x^2 \), we need to perform polynomial long division. The result will give us the quotient polynomial \( q(x) \) and the remainder polynomial \( r(x) \).

### Step 1: Perform Polynomial Long Division

We divide the highest degree term of \( a(x) \) by the highest degree term of \( b(x) \):

1. **First Division Step**:
   \[
   \frac{-7x^5}{x^2} = -7x^3
   \]
   Multiply \( -7x^3 \) by \( b(x) = x^2 \) and subtract from \( a(x) \):
   \[
   a(x) = -7x^5 + 3x^3 - 6x - 8
   \]
   \[
   (-7x^5 + 3x^3 - 6x - 8) - (-7x^5) = 3x^3 - 6x - 8
   \]

2. **Second Division Step**:
   \[
   \frac{3x^3}{x^2} = 3x
   \]
   Multiply \( 3x \) by \( b(x) = x^2 \) and subtract from the new expression:
   \[
   3x^3 - 6x - 8 - (3x^3) = -6x - 8
   \]

3. **Third Division Step**:
   \[
   \frac{-6x}{x^2} = -6/x \quad (\text{Not applicable for quotient, as degree would be negative})
   \]
   The remaining expression \( -6x - 8 \) is the remainder since its degree is less than that of \( b(x) \).

### Conclusion

The quotient \( q(x) \) is:
\[
q(x) = -7x^3 + 3x
\]
The remainder \( r(x) \) is:
\[
r(x) = -6x - 8
\]

### Final Answer:

- Quotient: \( q(x) = -7x^3 + 3x \)
- Remainder: \( r(x) = -6x - 8 \)

Would you like a detailed explanation of the steps? If you have any questions, feel free to ask!

Here are 8 related questions:

1. How do you perform polynomial long division step-by-step?
2. How can you check if your quotient and remainder are correct?
3. What happens when the divisor polynomial is of higher degree than the dividend?
4. Can you apply synthetic division to this problem?
5. What is the importance of the degree of the remainder in polynomial division?
6. How does the degree of the divisor affect the quotient and remainder?
7. What are some common mistakes to avoid in polynomial long division?
8. How would the process change if \( b(x) \) had a leading coefficient other than 1?

**Tip:** Always align the terms by degree when performing polynomial long division to avoid errors.

Learn how to perform polynomial long division to divide -7x^5 + 3x^3 - 6x - 8 by x^2. This detailed solution includes step-by-step instructions and explains the concepts of quotient and remainder in polynomial division. Suitable for high school students studying algebra.

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