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

Explanation

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

### Step 1: Expand the numerator
First, we need to calculate \((2 + 3i)^2\).

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

Since \(i^2 = -1\), we can substitute that in:

\[ = 4 + 12i + 9(-1) = 4 + 12i - 9 = -5 + 12i \]

### Step 2: Write the expression
Now we can rewrite the expression:

\[ z = \frac{-5 + 12i}{2 - i} \]

### Step 3: Rationalize the denominator
To simplify this expression, we multiply the numerator and denominator by the conjugate of the denominator, which is \(2 + i\):

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

### Step 4: Calculate the denominator
Now, calculate the denominator:

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

### Step 5: Calculate the numerator
Next, calculate the numerator:

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

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

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

### Step 6: Combine the results
Now, we can combine the results:

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

### Step 7: Write in the form \(a + ib\)
Thus, we have:

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

### Step 8: Find the conjugate
The conjugate of a complex number \(a + ib\) is \(a - ib\). Therefore, the conjugate of \(z\) is:

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

### Final Answer
So, the expression in the form \(a + ib\) is:

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

And the conjugate is:

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

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