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The problem provided asks to express the given complex number in standard form. The expression is:

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

### Step 1: Simplify each fraction

#### Fraction 1: \(\frac{1 - 2i}{2 + i}\)

To simplify, multiply both the numerator and denominator by the conjugate of the denominator \(2 - i\):

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

Simplify the numerator:
\[
2 - i - 4i - 2i^2 = 2 - 5i + 2 = 4 - 5i
\]
And the denominator:
\[
(2)^2 - (i)^2 = 4 - (-1) = 5
\]

So, the first fraction becomes:
\[
\frac{4 - 5i}{5} = \frac{4}{5} - \frac{5}{5}i = \frac{4}{5} - i
\]

#### Fraction 2: \(\frac{4 - i}{3 + 2i}\)

Similarly, multiply by the conjugate of the denominator \(3 - 2i\):

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

Simplify the numerator:
\[
12 - 8i - 3i - 2i^2 = 12 - 11i + 2 = 14 - 11i
\]
And the denominator:
\[
(3)^2 - (2i)^2 = 9 - 4(-1) = 9 + 4 = 13
\]

So, the second fraction becomes:
\[
\frac{14 - 11i}{13} = \frac{14}{13} - \frac{11}{13}i
\]

### Step 2: Add the two simplified fractions

Now add the two simplified results:

\[
\left( \frac{4}{5} - i \right) + \left( \frac{14}{13} - \frac{11}{13}i \right)
\]

Separate real and imaginary parts:

Real part:
\[
\frac{4}{5} + \frac{14}{13} = \frac{52}{65} + \frac{70}{65} = \frac{122}{65}
\]

Imaginary part:
\[
-1 - \frac{11}{13} = \frac{-13}{13} - \frac{11}{13} = \frac{-24}{13}
\]

Thus, the final result in standard form is:

\[
\frac{122}{65} - i\frac{24}{13}
\]

If you prefer, this can be approximated as:
\[
1.877 - 1.846i
\]

Would you like more details or have any questions?

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Here are 5 related questions to expand on this topic:
1. What is the significance of the conjugate in complex number division?
2. How do you calculate the magnitude of a complex number?
3. What are some applications of complex numbers in physics or engineering?
4. Can the standard form of a complex number be represented geometrically?
5. How do you add, subtract, multiply, and divide complex numbers?

**Tip:** When multiplying complex numbers, always remember that \(i^2 = -1\), which simplifies expressions quickly!

Express the following in standard form: (1 - 2i) / (2 + i) + (4 - i) / (3 + 2i)

Solve the problem of expressing (1 - 2i) / (2 + i) + (4 - i) / (3 + 2i) in standard form by using the conjugate multiplication method. This solution involves simplifying complex fractions and converting them into the standard form a + bi.

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