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The given problem asks to perform polynomial long division on \( -2x^3 - 8x^2 - 3x - 3 \) divided by \( -x - 4 \), and express the result in the form:

\[
q(x) + \frac{r(x)}{b(x)}
\]

Where:
- \( q(x) \) is the quotient,
- \( r(x) \) is the remainder,
- \( b(x) = -x - 4 \).

Let's perform the division step by step using polynomial long division.

1. Divide the leading term of the dividend \( -2x^3 \) by the leading term of the divisor \( -x \), which gives \( 2x^2 \).
   
2. Multiply \( 2x^2 \) by \( -x - 4 \), which gives \( -2x^3 - 8x^2 \).

3. Subtract \( (-2x^3 - 8x^2) \) from \( (-2x^3 - 8x^2 - 3x - 3) \), leaving \( 0x^3 + 0x^2 - 3x - 3 \), or simply \( -3x - 3 \).

4. Divide the leading term \( -3x \) by \( -x \), which gives \( 3 \).

5. Multiply \( 3 \) by \( -x - 4 \), which gives \( -3x - 12 \).

6. Subtract \( (-3x - 12) \) from \( (-3x - 3) \), leaving \( 9 \).

The quotient \( q(x) \) is \( 2x^2 + 3 \), and the remainder \( r(x) \) is \( 9 \).

Thus, the result of the division is:

\[
2x^2 + 3 + \frac{9}{-x - 4}
\]

Would you like further clarification or additional details on any of the steps? 

Here are five related questions you might find useful:
1. What is synthetic division, and how does it compare to polynomial long division?
2. How would the process change if we divided by \( x + 4 \) instead of \( -x - 4 \)?
3. What are the conditions for a polynomial division to have no remainder?
4. How can the remainder theorem help in checking the remainder quickly?
5. What are the uses of quotient and remainder in practical applications of polynomial division?

**Tip:** Polynomial long division follows the same logic as numerical long division; just treat each term of the polynomial like a number based on powers of \( x \).

Use the long division method to find the result when -2x^3 - 8x^2 - 3x - 3 is divided by -x - 4. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Learn how to divide the polynomial -2x^3 - 8x^2 - 3x - 3 by -x - 4 using long division. This step-by-step solution explains the quotient, remainder, and how to express the result in the form q(x) + r(x)/b(x). Suitable for Grades 10-12.

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