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The problem in the image asks to perform the indicated operations and write the complex number in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

The equation given is:

\[
2i + \frac{6 - i}{2 + 5i} = \_ + \_i
\]

### Step 1: Simplify the fraction \(\frac{6 - i}{2 + 5i}\)

We can multiply both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. The conjugate of \(2 + 5i\) is \(2 - 5i\).

\[
\frac{6 - i}{2 + 5i} \times \frac{2 - 5i}{2 - 5i} = \frac{(6 - i)(2 - 5i)}{(2 + 5i)(2 - 5i)}
\]

### Step 2: Multiply out both the numerator and denominator

#### Numerator:
\[
(6 - i)(2 - 5i) = 6 \cdot 2 + 6 \cdot (-5i) - i \cdot 2 - i \cdot (-5i)
= 12 - 30i - 2i + 5i^2
\]

Since \(i^2 = -1\), we replace \(5i^2\) with \(-5\):
\[
12 - 30i - 2i - 5 = 7 - 32i
\]

#### Denominator:
\[
(2 + 5i)(2 - 5i) = 2^2 - (5i)^2 = 4 - 25i^2 = 4 + 25 = 29
\]

### Step 3: Simplify the fraction
\[
\frac{7 - 32i}{29} = \frac{7}{29} - \frac{32}{29}i
\]

### Step 4: Add this result to \(2i\)

Now add \(2i\) to the result:
\[
2i + \left(\frac{7}{29} - \frac{32}{29}i\right) = \frac{7}{29} + \left(2 - \frac{32}{29}\right)i
\]

Simplifying the imaginary part:
\[
2 - \frac{32}{29} = \frac{58}{29} - \frac{32}{29} = \frac{26}{29}
\]

### Final result:
\[
\frac{7}{29} + \frac{26}{29}i
\]

Thus, the complex number in standard form is:

\[
\boxed{\frac{7}{29} + \frac{26}{29}i}
\]

Would you like further clarification on any step, or have questions?

### Related questions:
1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate in complex division?
3. Why does \(i^2 = -1\) in complex numbers?
4. How would the result change if the real parts were different?
5. How do you interpret the real and imaginary parts geometrically?

### Tip:
When simplifying complex fractions, always multiply by the conjugate to eliminate imaginary terms from the denominator.

Perform the indicated operations and write the complex number in the standard form a + bi. 2i + (6 - i)/(2 + 5i) = ___ + ___i.

Learn how to simplify complex number expressions such as 2i + (6 - i)/(2 + 5i) and write them in the standard form a + bi. This detailed solution walks through multiplying by the conjugate and simplifying the real and imaginary components. Suitable for Grades 9-12.

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