Convert `((2+3i)^(2))/(2-i)` in the form of `a+ib` and find its conjugate.

Explanation

To convert \(\frac{(2 + 3i)^2}{2 - i}\) into the form \(a + ib\) and find its conjugate, we will follow these steps:

### Step 1: Calculate \((2 + 3i)^2\)

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\):

\[ (2 + 3i)^2 = 2^2 + 2 \cdot 2 \cdot 3i + (3i)^2 \]
\[ = 4 + 12i + 9i^2 \]

Since \(i^2 = -1\):

\[ = 4 + 12i - 9 = -5 + 12i \]

### Step 2: Substitute back into the expression

Now, substitute \((2 + 3i)^2\) back into the expression:

\[ \frac{(2 + 3i)^2}{2 - i} = \frac{-5 + 12i}{2 - i} \]

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

To simplify \(\frac{-5 + 12i}{2 - i}\), we multiply the numerator and the denominator by the conjugate of the denominator, which is \(2 + i\):

\[ \frac{(-5 + 12i)(2 + i)}{(2 - i)(2 + i)} \]

### Step 4: Calculate the denominator

Calculate the denominator:

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

### Step 5: Calculate the numerator

Now calculate the numerator:

\[ (-5 + 12i)(2 + i) = -5 \cdot 2 + (-5) \cdot i + 12i \cdot 2 + 12i \cdot i \]
\[ = -10 - 5i + 24i + 12i^2 \]
\[ = -10 + 19i + 12(-1) = -10 + 19i - 12 = -22 + 19i \]

### Step 6: Combine the results

Now we can combine the results:

\[ \frac{-22 + 19i}{5} = -\frac{22}{5} + \frac{19}{5}i \]

Thus, in the form \(a + ib\):

\[ a = -\frac{22}{5}, \quad b = \frac{19}{5} \]

### Step 7: Find the conjugate

The conjugate of a complex number \(a + ib\) is \(a - ib\). Therefore, the conjugate of \(-\frac{22}{5} + \frac{19}{5}i\) is:

\[ -\frac{22}{5} - \frac{19}{5}i \]

### Final Answer

The expression \(\frac{(2 + 3i)^2}{2 - i}\) in the form \(a + ib\) is:

\[ -\frac{22}{5} + \frac{19}{5}i \]

And its conjugate is:

\[ -\frac{22}{5} - \frac{19}{5}i \]