Express `(1- 2i)/(2+i) + (3+i)/(2-i)` in the form `a + bi`

Answer

To express \(\frac{1 - 2i}{2 + i} + \frac{3 + i}{2 - i}\) in the form \(a + bi\), we will follow these steps:

### Step 1: Find a common denominator
We need to combine the two fractions. The common denominator will be the product of the two denominators: \((2 + i)(2 - i)\).

### Step 2: Rewrite the expression with the common denominator
The expression can be rewritten as:
\[ \frac{(1 - 2i)(2 - i) + (3 + i)(2 + i)}{(2 + i)(2 - i)} \]

### Step 3: Calculate the denominator
Now, we calculate the denominator:
\[ (2 + i)(2 - i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

### Step 4: Calculate the numerator
Next, we calculate the numerator:
1. For \((1 - 2i)(2 - i)\):
\[ = 1 \cdot 2 + 1 \cdot (-i) - 2i \cdot 2 - 2i \cdot (-i) = 2 - i - 4i + 2i^2 \]
Since \(i^2 = -1\), we have:
\[ 2 - i - 4i - 2 = 0 - 5i = -5i \]

2. For \((3 + i)(2 + i)\):
\[ = 3 \cdot 2 + 3 \cdot i + i \cdot 2 + i \cdot i = 6 + 3i + 2i + i^2 \]
Again using \(i^2 = -1\):
\[ 6 + 5i - 1 = 5 + 5i \]

Now, combine the results from both parts of the numerator:
\[ -5i + (5 + 5i) = 5 + 0i = 5 \]

### Step 5: Combine the results
Now, we can combine the results:
\[ \frac{5}{5} = 1 \]

### Final Result
Thus, we express the original expression in the form \(a + bi\):
\[ 1 + 0i \]