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

Answer

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

### Step 1: Expand the numerator \((3-i)^2\)

Using the formula for the square of a binomial, \((a-b)^2 = a^2 - 2ab + b^2\), we can expand \((3-i)^2\):

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

Calculating this gives:

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

### Step 2: Rewrite the expression

Now we can rewrite the expression as:

\[ \frac{8 - 6i}{2 + i} \]

### Step 3: Multiply by the conjugate of the denominator

To eliminate the imaginary part in the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \(2 - i\):

\[ \frac{(8 - 6i)(2 - i)}{(2 + i)(2 - i)} \]

### Step 4: Calculate the denominator

Calculating the denominator:

\[ (2 + i)(2 - i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

### Step 5: Calculate the numerator

Now, we calculate the numerator:

\[ (8 - 6i)(2 - i) = 8 \cdot 2 - 8 \cdot i - 6i \cdot 2 + 6i \cdot i \]
\[ = 16 - 8i - 12i + 6(-1) = 16 - 8i - 12i - 6 = 16 - 6 - 20i = 10 - 20i \]

### Step 6: Combine the results

Now we can combine the results:

\[ \frac{10 - 20i}{5} \]

### Step 7: Simplify the expression

Dividing both the real and imaginary parts by 5 gives:

\[ = \frac{10}{5} - \frac{20i}{5} = 2 - 4i \]

### Final Answer

Thus, the expression \(\frac{(3-i)^2}{2+i}\) in the form \(a + bi\) is:

\[ \boxed{2 - 4i} \]