(X - 3I) (3X + 5I ) – (3X + 5I) (X + 2I )

We are given the expression:

\[
(X - 3I)(3X + 5I) - (3X + 5I)(X + 2I)
\]

### Step 1: Expand the first term
Using the distributive property \((A+B)(C+D) = AC + AD + BC + BD\), expand the first term \((X - 3I)(3X + 5I)\):

\[
(X - 3I)(3X + 5I) = X(3X) + X(5I) - 3I(3X) - 3I(5I)
\]

Simplify each term:

\[
= 3X^2 + 5XI - 9XI - 15I^2
\]

Since \(I^2 = -1\), this becomes:

\[
3X^2 + 5XI - 9XI + 15 = 3X^2 - 4XI + 15
\]

### Step 2: Expand the second term
Now expand \((3X + 5I)(X + 2I)\):

\[
(3X + 5I)(X + 2I) = 3X(X) + 3X(2I) + 5I(X) + 5I(2I)
\]

Simplify each term:

\[
= 3X^2 + 6XI + 5XI + 10I^2
\]

Since \(I^2 = -1\), this becomes:

\[
= 3X^2 + 6XI + 5XI - 10 = 3X^2 + 11XI - 10
\]

### Step 3: Subtract the two results
Now subtract the expanded second term from the first term:

\[
(3X^2 - 4XI + 15) - (3X^2 + 11XI - 10)
\]

Distribute the negative sign:

\[
3X^2 - 4XI + 15 - 3X^2 - 11XI + 10
\]

Simplify by combining like terms:

\[
0 - 15XI + 25 = -15XI + 25
\]

### Final Answer:
The simplified expression is:

\[
-15XI + 25
\]

Let me know if you need more details or have any questions! Here are 5 related questions to deepen your understanding:

1. How do you apply the distributive property in more complex algebraic expressions?
2. How do imaginary numbers affect the result of polynomial expansions?
3. What are some common mistakes when simplifying expressions involving complex numbers?
4. How can we verify the correctness of expanded polynomial expressions?
5. What is the significance of \( I^2 = -1 \) in these calculations?

**Tip:** Always remember that \( I^2 = -1 \) when simplifying expressions with imaginary numbers.

Learn how to simplify the expression (X - 3I)(3X + 5I) – (3X + 5I)(X + 2I) using polynomial expansion and properties of imaginary numbers. This step-by-step solution covers key concepts in algebra and complex numbers, making it suitable for students in Grades 9-12.

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