Certainly! To solve the problem using synthetic division, we'll focus on simplifying the fraction \(\frac{4x^2 + 10x + 4}{x^2(2x + 1)}\) by first using synthetic division.

Since synthetic division is typically used when dividing by a linear factor, we will apply it here in the following steps:

### Step 1: Division of \(4x^2 + 10x + 4\) by \(2x + 1\)

Since the divisor is \(2x + 1\), we can start by performing synthetic division on \(4x^2 + 10x + 4\).

1. **Divide the leading coefficient of the numerator by the leading coefficient of the divisor:**
   - Here, the divisor's leading term is \(2x\), and the numerator's leading term is \(4x^2\).
   - Divide \(4x^2\) by \(2x\), which gives \(2x\).

2. **Multiply the divisor \(2x + 1\) by the quotient \(2x\):**
   \[
   2x(2x + 1) = 4x^2 + 2x
   \]

3. **Subtract this from the original polynomial:**
   \[
   (4x^2 + 10x + 4) - (4x^2 + 2x) = 8x + 4
   \]

### Step 2: Divide the remainder \(8x + 4\) by \(2x + 1\)

1. **Divide the leading term \(8x\) by \(2x\), which gives \(4\).**

2. **Multiply the entire divisor by this quotient:**
   \[
   4(2x + 1) = 8x + 4
   \]

3. **Subtract to get the remainder:**
   \[
   (8x + 4) - (8x + 4) = 0
   \]

### Step 3: Write the result

The quotient from synthetic division is:
\[
2x + 4
\]

### Final Result:

Given that the original expression was \(\frac{4x^2 + 10x + 4}{x^2(2x + 1)}\), we have simplified it to:
\[
\frac{2x + 4}{x^2}
\]

This can now be written as:
\[
\frac{2x + 4}{x^2} = \frac{2x}{x^2} + \frac{4}{x^2} = \frac{2}{x} + \frac{4}{x^2}
\]

Thus, the partial fraction decomposition of the original fraction is:
\[
\frac{2}{x} + \frac{4}{x^2}
\]

---

Do you have any more questions or would you like more details? Here are five related questions:

1. How does synthetic division differ from long division?
2. Can synthetic division be used when the divisor is not linear?
3. What are the advantages of using synthetic division over other methods?
4. How do you handle synthetic division when the divisor has a leading coefficient other than 1?
5. How would the process change if we were dividing by a quadratic polynomial instead of a linear one?

**Tip:** When using synthetic division, ensure that all powers of \(x\) are accounted for, including any that might be missing, by inserting coefficients of zero where needed.

Learn how to use synthetic division to simplify the fraction 4x^2 + 10x + 4 divided by x^2(2x + 1). This detailed step-by-step guide covers synthetic division algorithm, polynomial division, and partial fraction decomposition. Suitable for high school students in Grades 10-12.

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