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To simplify the expression \(\frac{5 + 3i}{4 - 2i}\), let's go through the steps to find the correct answer.

### Step 1: Multiply by the Conjugate
To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, \(4 + 2i\):

\[
\frac{5 + 3i}{4 - 2i} \cdot \frac{4 + 2i}{4 + 2i} = \frac{(5 + 3i)(4 + 2i)}{(4 - 2i)(4 + 2i)}
\]

### Step 2: Expand the Numerator and Denominator
#### Numerator:
\[
(5 + 3i)(4 + 2i) = 5 \cdot 4 + 5 \cdot 2i + 3i \cdot 4 + 3i \cdot 2i
\]
\[
= 20 + 10i + 12i + 6i^2
\]
Since \(i^2 = -1\), we substitute \(6i^2\) with \(-6\):
\[
= 20 + 10i + 12i - 6 = 14 + 22i
\]

#### Denominator:
\[
(4 - 2i)(4 + 2i) = 4^2 - (2i)^2 = 16 - 4i^2
\]
Since \(i^2 = -1\), we replace \(4i^2\) with \(-4\):
\[
= 16 + 4 = 20
\]

### Step 3: Simplify the Expression
Now we have:
\[
\frac{14 + 22i}{20} = \frac{14}{20} + \frac{22}{20}i
\]
Simplify each part:
\[
= \frac{7}{10} + \frac{11}{10}i
\]

### Conclusion
The simplified form is \(\frac{7}{10} + \frac{11}{10}i\), which corresponds to option **A**.

Would you like further clarification or additional details on any step?

Here are some related questions for further practice:
1. What is the conjugate of a complex number and why is it useful in simplification?
2. How do you multiply complex numbers and handle terms with \(i^2\)?
3. Can you simplify \(\frac{3 + 4i}{2 - i}\) using similar steps?
4. What are some common mistakes to avoid when simplifying complex fractions?
5. How does simplifying complex numbers relate to real-world applications?

**Tip:** Always multiply both the numerator and denominator by the conjugate of the denominator to simplify complex fractions.

Learn how to simplify the complex fraction (5 + 3i) / (4 - 2i) by multiplying by the conjugate. This solution covers rationalizing the denominator, using complex number rules, and handling imaginary units. Suitable for high school students in Grades 9-12.

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