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Let's simplify the given expression and obtain the binomial form:

\[
\frac{3 - 2i}{3 + 4i} + \frac{3 + i}{3 - i}
\]

### Step 1: Simplify \(\frac{3 - 2i}{3 + 4i}\)
To simplify \(\frac{3 - 2i}{3 + 4i}\), multiply both the numerator and the denominator by the conjugate of the denominator \(3 - 4i\):

\[
\frac{3 - 2i}{3 + 4i} \times \frac{3 - 4i}{3 - 4i} = \frac{(3 - 2i)(3 - 4i)}{(3 + 4i)(3 - 4i)}
\]

First, calculate the denominator:

\[
(3 + 4i)(3 - 4i) = 9 - 16i^2 = 9 + 16 = 25
\]

Now, expand the numerator:

\[
(3 - 2i)(3 - 4i) = 9 - 12i - 6i + 8i^2 = 9 - 18i - 8 = 1 - 18i
\]

So:

\[
\frac{3 - 2i}{3 + 4i} = \frac{1 - 18i}{25} = \frac{1}{25} - \frac{18}{25}i
\]

### Step 2: Simplify \(\frac{3 + i}{3 - i}\)
To simplify \(\frac{3 + i}{3 - i}\), multiply both the numerator and the denominator by the conjugate of the denominator \(3 + i\):

\[
\frac{3 + i}{3 - i} \times \frac{3 + i}{3 + i} = \frac{(3 + i)(3 + i)}{(3 - i)(3 + i)}
\]

First, calculate the denominator:

\[
(3 - i)(3 + i) = 9 - i^2 = 9 + 1 = 10
\]

Now, expand the numerator:

\[
(3 + i)(3 + i) = 9 + 6i + i^2 = 9 + 6i - 1 = 8 + 6i
\]

So:

\[
\frac{3 + i}{3 - i} = \frac{8 + 6i}{10} = \frac{8}{10} + \frac{6}{10}i = 0.8 + 0.6i
\]

### Step 3: Combine the Results
Now, combine the two simplified terms:

\[
\frac{1}{25} - \frac{18}{25}i + 0.8 + 0.6i
\]

Combine the real parts and the imaginary parts:

\[
\left(\frac{1}{25} + 0.8\right) + \left(-\frac{18}{25} + 0.6\right)i
\]

\[
\text{Real part: } \frac{1}{25} + \frac{20}{25} = \frac{21}{25} = 0.84
\]

\[
\text{Imaginary part: } -\frac{18}{25} + \frac{15}{25} = -\frac{3}{25} = -0.12
\]

### Final Answer:
The binomial form of the expression is:

\[
\boxed{0.84 - 0.12i}
\]

Would you like further explanation or have any additional questions?

### Related Questions:
1. How do you find the conjugate of a complex number?
2. Why is it important to remove the imaginary unit from the denominator?
3. How do you add or subtract complex numbers?
4. What is the geometric interpretation of multiplying complex numbers?
5. How can complex fractions be simplified more efficiently?

### Tip:
When simplifying complex fractions, always remember to multiply by the conjugate to clear the imaginary part from the denominator. This is a key step to ensure the expression is in a proper binomial form.

Learn how to simplify complex fractions involving complex numbers using conjugates. This detailed guide covers the step-by-step process with examples and explanations, suitable for advanced high school and college students.

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