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The image shows the expression \(\frac{2x^2 + x}{4x^3 + 0x^2 + 10x + 4}\), and you want to use synthetic division to simplify or solve it.

### Steps to Use Synthetic Division

1. **Identify the divisor and dividend**:
   - The **dividend** is \(4x^3 + 0x^2 + 10x + 4\).
   - The **divisor** is \(2x^2 + x\).

   However, synthetic division is usually used when dividing a polynomial by a linear expression (e.g., \(x - c\)). In this case, because the divisor is quadratic, the synthetic division approach needs to be adapted, or traditional polynomial division should be used instead. 

2. **Traditional Polynomial Division Setup**:
   - Divide \(4x^3 + 0x^2 + 10x + 4\) by \(2x^2 + x\).

### Polynomial Division Process

1. **Divide the leading term** of the dividend \(4x^3\) by the leading term of the divisor \(2x^2\):
   \[
   \frac{4x^3}{2x^2} = 2x
   \]
   This is the first term of the quotient.

2. **Multiply** the entire divisor by \(2x\) and subtract this from the dividend:
   \[
   2x \cdot (2x^2 + x) = 4x^3 + 2x^2
   \]
   Subtract from the dividend:
   \[
   (4x^3 + 0x^2 + 10x + 4) - (4x^3 + 2x^2) = -2x^2 + 10x + 4
   \]

3. **Repeat** the process with the new polynomial \(-2x^2 + 10x + 4\):
   - Divide \(-2x^2\) by \(2x^2\):
   \[
   \frac{-2x^2}{2x^2} = -1
   \]
   - Multiply the divisor by \(-1\):
   \[
   -1 \cdot (2x^2 + x) = -2x^2 - x
   \]
   - Subtract from the new dividend:
   \[
   (-2x^2 + 10x + 4) - (-2x^2 - x) = 11x + 4
   \]

4. **Conclusion**:
   - The quotient is \(2x - 1\), and the remainder is \(11x + 4\).

### Final Answer
\[
\frac{4x^3 + 10x + 4}{2x^2 + x} = 2x - 1 + \frac{11x + 4}{2x^2 + x}
\]

This result means that \(2x - 1\) is the quotient, and \(\frac{11x + 4}{2x^2 + x}\) is the remainder in fractional form.

Would you like more details on any part of this process or further explanation?

### Questions to Explore Further:
1. What are the conditions under which synthetic division is applicable?
2. How does synthetic division differ from long polynomial division?
3. What are other methods for dividing polynomials?
4. Can the remainder be further simplified?
5. How would the process change if the divisor were a linear polynomial?

### Tip:
When using synthetic division, ensure the divisor is of the form \(x - c\). For higher-degree divisors, traditional polynomial division or other methods are more appropriate.

Explore how to use synthetic division to simplify or solve complex polynomial expressions. This detailed guide covers the steps involved in using synthetic division with examples. Suitable for advanced high school students and above.

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