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The problem asks to find the real and imaginary parts of several expressions involving complex numbers \( w \) and \( x \), where:
\[
w = -3 + i \quad \text{and} \quad x = 2 + 3i.
\]

Let's analyze each expression and find the real and imaginary parts:

1. **\( w + x \)**:
   \[
   w + x = (-3 + i) + (2 + 3i) = (-3 + 2) + (i + 3i) = -1 + 4i.
   \]
   - **Real part:** \(-1\)
   - **Imaginary part:** \(4\)

2. **\( w - x \)**:
   \[
   w - x = (-3 + i) - (2 + 3i) = (-3 - 2) + (i - 3i) = -5 - 2i.
   \]
   - **Real part:** \(-5\)
   - **Imaginary part:** \(-2\)

3. **\( wx \)** (multiplication):
   \[
   wx = (-3 + i)(2 + 3i).
   \]
   Using the distributive property:
   \[
   wx = (-3 \cdot 2) + (-3 \cdot 3i) + (i \cdot 2) + (i \cdot 3i) = -6 - 9i + 2i + 3i^2.
   \]
   Since \( i^2 = -1 \), this becomes:
   \[
   wx = -6 - 9i + 2i + 3(-1) = -6 - 7i - 3 = -9 - 7i.
   \]
   - **Real part:** \(-9\)
   - **Imaginary part:** \(-7\)

4. **\( \frac{w}{x} \)** (division):
   \[
   \frac{w}{x} = \frac{-3 + i}{2 + 3i}.
   \]
   To simplify, we multiply the numerator and denominator by the complex conjugate of the denominator:
   \[
   \frac{(-3 + i)(2 - 3i)}{(2 + 3i)(2 - 3i)}.
   \]
   The denominator simplifies to:
   \[
   (2 + 3i)(2 - 3i) = 2^2 - (3i)^2 = 4 - 9(-1) = 4 + 9 = 13.
   \]
   Now, let's simplify the numerator:
   \[
   (-3 + i)(2 - 3i) = (-3 \cdot 2) + (-3 \cdot -3i) + (i \cdot 2) - (i \cdot 3i) = -6 + 9i + 2i - 3(-1).
   \]
   \[
   = -6 + 11i + 3 = -3 + 11i.
   \]
   So:
   \[
   \frac{w}{x} = \frac{-3 + 11i}{13} = -\frac{3}{13} + \frac{11}{13}i.
   \]
   - **Real part:** \(-\frac{3}{13}\)
   - **Imaginary part:** \(\frac{11}{13}\)

Would you like any additional details or explanations for these calculations? 

Here are 5 related questions to expand your understanding:
1. What is the general formula for multiplying two complex numbers?
2. How can you find the complex conjugate of a given complex number?
3. Why do we multiply by the conjugate when dividing complex numbers?
4. What happens when you add or subtract two purely imaginary numbers?
5. Can a real number be considered a complex number? If so, how?

**Tip:** When dividing complex numbers, always multiply by the conjugate of the denominator to simplify the expression.

Find the real and imaginary parts of the following, given w = -3 + i and x = 2 + 3i:

Learn how to find the real and imaginary parts of complex number expressions, including addition, subtraction, multiplication, and division. The complex numbers involved are w = -3 + i and x = 2 + 3i. Suitable for students in Grades 10-12.

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