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The expression to simplify is:

\[
\frac{1 + 2i}{3 + i}
\]

To simplify this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 + i\) is \(3 - i\).

\[
\frac{1 + 2i}{3 + i} \times \frac{3 - i}{3 - i} = \frac{(1 + 2i)(3 - i)}{(3 + i)(3 - i)}
\]

First, simplify the denominator:

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

Now simplify the numerator:

\[
(1 + 2i)(3 - i) = 1 \cdot 3 + 1 \cdot (-i) + 2i \cdot 3 + 2i \cdot (-i)
\]
\[
= 3 - i + 6i - 2i^2
\]
\[
= 3 + 5i - 2(-1) = 3 + 5i + 2 = 5 + 5i
\]

Thus, the expression becomes:

\[
\frac{5 + 5i}{10} = \frac{5}{10} + \frac{5i}{10} = \frac{1}{2} + \frac{1}{2}i
\]

So the simplified form is:

\[
\frac{1}{2} + \frac{1}{2}i
\]

If you need more details or have any questions, feel free to ask! Here are five related questions you can explore:

1. What is the general form of a complex number?
2. How do you find the conjugate of a complex number?
3. Why does multiplying by the conjugate help simplify complex fractions?
4. What are some other methods for simplifying complex numbers?
5. Can you apply this method to different types of complex fractions?

**Tip:** Always multiply by the conjugate when simplifying complex fractions; this removes the imaginary part from the denominator.

Learn how to simplify the complex fraction (1 + 2i) / (3 + i) by multiplying by the conjugate. This problem covers key concepts in complex numbers and algebraic fractions, including using conjugates and simplifying imaginary numbers. Suitable for students in Grades 10-12.

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