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To solve the given problem using synthetic division, we are tasked with dividing:

\[
\frac{4x^3 - 12x^2 - 5x - 9}{2x + 1}
\]

However, synthetic division is typically used for division by binomials of the form \(x - c\), not for a term with a coefficient like \(2x + 1\). So, we first need to rewrite \(2x + 1\) in a suitable form for synthetic division.

1. **Step 1: Transform the divisor**  
   Divide both the numerator and the divisor by 2. This gives us:

\[
\frac{4x^3 - 12x^2 - 5x - 9}{2x + 1} = \frac{2(2x^3 - 6x^2 - \frac{5}{2}x - \frac{9}{2})}{2(2x + 1)}
\]

Thus, we rewrite it as:

\[
\frac{2x^3 - 6x^2 - \frac{5}{2}x - \frac{9}{2}}{x + \frac{1}{2}}
\]

Now, we can proceed with synthetic division by dividing by \(x + \frac{1}{2}\).

2. **Step 2: Set up the synthetic division**  
We use \(x = -\frac{1}{2}\) for synthetic division. The coefficients of the numerator are \(2\), \(-6\), \(-\frac{5}{2}\), and \(-\frac{9}{2}\).

| Coefficients | 2 | -6 | -5/2 | -9/2 |
|--------------|---|----|------|------|
| \(x = -\frac{1}{2}\) |

**Synthetic division process**:
1. Bring down the first coefficient: 2.
2. Multiply 2 by \(-\frac{1}{2}\) and add to \(-6\):  
   \(2 \times -\frac{1}{2} = -1\), and \(-6 + (-1) = -7\).
3. Multiply \(-7\) by \(-\frac{1}{2}\) and add to \(-\frac{5}{2}\):  
   \(-7 \times -\frac{1}{2} = \frac{7}{2}\), and \(-\frac{5}{2} + \frac{7}{2} = 1\).
4. Multiply 1 by \(-\frac{1}{2}\) and add to \(-\frac{9}{2}\):  
   \(1 \times -\frac{1}{2} = -\frac{1}{2}\), and \(-\frac{9}{2} + (-\frac{1}{2}) = -5\).

The quotient is \(2x^2 - 7x + 1\) with a remainder of \(-5\).

Thus, the result of the division is:

\[
2x^2 - 7x + 1 - \frac{5}{2x + 1}
\]

Would you like further clarification or a detailed explanation of any steps?

Here are 5 related questions:
1. How do you apply synthetic division to higher-degree polynomials?
2. What is the importance of transforming the divisor in synthetic division?
3. Can synthetic division be used for non-binomial divisors? Why or why not?
4. What are the alternative methods to divide polynomials if synthetic division is not applicable?
5. How does the remainder theorem relate to synthetic division?

**Tip**: Synthetic division is faster than long division, but only works when dividing by linear binomials of the form \(x - c\).

Use synthetic division to divide (4x^3 - 12x^2 - 5x - 9) ÷ (2x + 1).

Learn how to use synthetic division to divide the polynomial (4x^3 - 12x^2 - 5x - 9) by (2x + 1). This problem covers algebraic concepts, including polynomial division and the remainder theorem, and is suitable for students in Grades 10-12.

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