Let's simplify the expression \( \frac{(5 + J^2)(4 - 5J)}{2 + 3J} \), where \( J \) represents the imaginary unit, i.e., \( J = \sqrt{-1} \).

### Step 1: Expand the numerator
The numerator is \( (5 + J^2)(4 - 5J) \).

First, recall that \( J^2 = -1 \). So:
\[
5 + J^2 = 5 - 1 = 4
\]

Now multiply \( 4 \) by \( (4 - 5J) \):
\[
4(4 - 5J) = 16 - 20J
\]

Thus, the numerator simplifies to \( 16 - 20J \).

### Step 2: Express the denominator in standard form
The denominator is \( 2 + 3J \). To simplify the division, we multiply both the numerator and the denominator by the complex conjugate of the denominator, \( 2 - 3J \).

### Step 3: Multiply the numerator and denominator by the complex conjugate
\[
\frac{(16 - 20J)(2 - 3J)}{(2 + 3J)(2 - 3J)}
\]

#### Step 3.1: Multiply the denominator
\[
(2 + 3J)(2 - 3J) = 2^2 - (3J)^2 = 4 - (-9) = 4 + 9 = 13
\]

#### Step 3.2: Multiply the numerator
Now, expand \( (16 - 20J)(2 - 3J) \):
\[
16(2 - 3J) - 20J(2 - 3J) = 32 - 48J - 40J + 60J^2
\]

Since \( J^2 = -1 \), we get:
\[
32 - 48J - 40J + 60(-1) = 32 - 48J - 40J - 60 = -28 - 88J
\]

### Step 4: Final simplification
Thus, the expression becomes:
\[
\frac{-28 - 88J}{13}
\]
This simplifies to:
\[
\frac{-28}{13} + \frac{-88J}{13} = -\frac{28}{13} - \frac{88}{13}J
\]

So, the simplified result is:
\[
-\frac{28}{13} - \frac{88}{13}J
\]

Would you like more details or have any questions?

Here are some related questions:
1. How can we use polar form to simplify division of complex numbers?
2. What is the significance of multiplying by the complex conjugate in complex number division?
3. How do you convert a complex number from rectangular form to polar form?
4. How would the steps differ if the denominator involved a higher power of \( J \)?
5. What is the geometric interpretation of dividing complex numbers?

**Tip**: Always check for opportunities to use the complex conjugate when dividing complex numbers to avoid imaginary terms in the denominator.

Learn how to simplify complex expressions like (5 + J²)(4 - 5J)/(2 + 3J) step-by-step. This guide covers key concepts such as multiplying complex numbers, using the complex conjugate, and simplifying expressions involving imaginary units (J² = -1). Suitable for Grades 11-12 and early college students.

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