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We need to simplify the given complex fraction \(\frac{2+i}{5+2i}\) and write it in standard form, which is \(a + bi\), where \(a\) and \(b\) are real numbers.

To do this, we multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

1. **Given expression**:
   \[
   \frac{2+i}{5+2i}
   \]
   
2. **Multiply by the conjugate of the denominator**:
   The conjugate of \(5 + 2i\) is \(5 - 2i\). So multiply both the numerator and the denominator by \(5 - 2i\):
   \[
   \frac{(2+i)(5-2i)}{(5+2i)(5-2i)}
   \]

3. **Simplify the denominator**:
   \[
   (5+2i)(5-2i) = 5^2 - (2i)^2 = 25 - (-4) = 29
   \]

4. **Simplify the numerator**:
   Now multiply out \((2 + i)(5 - 2i)\):
   \[
   (2+i)(5-2i) = 2(5) + 2(-2i) + i(5) + i(-2i)
   \]
   \[
   = 10 - 4i + 5i - 2i^2
   \]
   Since \(i^2 = -1\):
   \[
   = 10 - 4i + 5i + 2 = 12 + i
   \]

5. **Final result**:
   The fraction becomes:
   \[
   \frac{12 + i}{29} = \frac{12}{29} + \frac{1}{29}i
   \]

Thus, the simplified form is \(\frac{12}{29} + \frac{1}{29}i\), which corresponds to option **C**.

Would you like further clarification on any steps?

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**Related Questions:**
1. What is the conjugate of a complex number?
2. How do you simplify complex fractions?
3. Why does multiplying by the conjugate remove the imaginary part?
4. What are the steps for multiplying complex numbers?
5. How is standard form for complex numbers defined?

**Tip:** Multiplying by the conjugate of a complex number is a crucial technique for simplifying complex fractions.

Simplify (2+i) / (5+2i) and write the answer in standard form.

Learn how to simplify the complex fraction (2+i)/(5+2i) by multiplying by the conjugate and converting to standard form. This problem demonstrates key concepts in algebra and complex numbers, including the conjugate theorem.

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