Find the complex conjugate fo the following
`(3 +4i) (2 - 3i)`

Answer

To find the complex conjugate of the expression \((3 + 4i)(2 - 3i)\), we will follow these steps:

### Step 1: Multiply the complex numbers
We start by multiplying the two complex numbers:

\[ (3 + 4i)(2 - 3i) \]

Using the distributive property (also known as the FOIL method for binomials):

\[ = 3 \cdot 2 + 3 \cdot (-3i) + 4i \cdot 2 + 4i \cdot (-3i) \]

Calculating each term:

\[ = 6 - 9i + 8i - 12i^2 \]

### Step 2: Simplify the expression
Now, we know that \(i^2 = -1\). Therefore, we can substitute \(-1\) for \(i^2\):

\[ = 6 - 9i + 8i + 12 \]

Now, combine the real parts and the imaginary parts:

\[ = (6 + 12) + (-9i + 8i) \]

This simplifies to:

\[ = 18 - i \]

### Step 3: Find the complex conjugate
The complex number we obtained is \(18 - i\). The complex conjugate of a complex number \(z = x + yi\) is given by \(z^* = x - yi\).

In our case:

\[ z = 18 - i \implies z^* = 18 + i \]

### Final Answer
Thus, the complex conjugate of \((3 + 4i)(2 - 3i)\) is:

\[ \boxed{18 + i} \]

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