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The problem asks to perform long division on the polynomial \( 3x^3 - 24x^2 - 4x + 29 \) divided by \( x - 8 \), and express the result in the form:

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

where \( q(x) \) is the quotient and \( r(x) \) is the remainder after division. Let's solve this using the long division method.

### Step-by-Step Process:

1. **Divide the leading term:**
   Divide the leading term \( 3x^3 \) by the leading term of \( x - 8 \), which is \( x \). This gives:

   \[
   \frac{3x^3}{x} = 3x^2
   \]

2. **Multiply and subtract:**
   Multiply \( 3x^2 \) by \( x - 8 \), which gives:

   \[
   3x^2(x - 8) = 3x^3 - 24x^2
   \]

   Now, subtract \( (3x^3 - 24x^2) \) from \( (3x^3 - 24x^2 - 4x + 29) \):

   \[
   (3x^3 - 24x^2 - 4x + 29) - (3x^3 - 24x^2) = -4x + 29
   \]

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

   \[
   \frac{-4x}{x} = -4
   \]

4. **Multiply and subtract:**
   Multiply \( -4 \) by \( x - 8 \), which gives:

   \[
   -4(x - 8) = -4x + 32
   \]

   Subtract \( (-4x + 32) \) from \( (-4x + 29) \):

   \[
   (-4x + 29) - (-4x + 32) = 29 - 32 = -3
   \]

### Final Result:
The quotient is \( 3x^2 - 4 \) and the remainder is \( -3 \). Thus, the result can be written as:

\[
q(x) = 3x^2 - 4, \quad r(x) = -3, \quad b(x) = x - 8
\]

So the final expression is:

\[
3x^2 - 4 + \frac{-3}{x - 8}
\]

Would you like further clarification on any step or more details? Here are five related questions:

1. How do you handle polynomial division if the degree of the divisor is higher?
2. What happens if the remainder in long division is zero?
3. Can synthetic division be used for this type of problem?
4. What is the relationship between long division of polynomials and factorization?
5. How can you verify the result of polynomial division?

**Tip:** Always ensure that the polynomial is written in standard form (with terms arranged in descending powers) before starting long division.

Use the long division method to find the result when 3x^3 - 24x^2 - 4x + 29 is divided by x - 8. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Learn how to perform polynomial long division by dividing 3x^3 - 24x^2 - 4x + 29 by x - 8. This step-by-step solution explains the process and provides a clear result, including the quotient and remainder. Suitable for high school students in grades 9-12.

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