To solve \((2 - 3i) \times (3 + i)\), we use the distributive property (also known as the FOIL method for binomials).

Here's the step-by-step solution:

1. **First terms**: \(2 \times 3 = 6\)
2. **Outer terms**: \(2 \times i = 2i\)
3. **Inner terms**: \(-3i \times 3 = -9i\)
4. **Last terms**: \(-3i \times i = -3i^2\)

Now, we combine these results:

\[ 6 + 2i - 9i - 3i^2 \]

Next, simplify the imaginary parts and remember that \(i^2 = -1\):

\[ 6 + 2i - 9i - 3(-1) \]

\[ 6 + 2i - 9i + 3 \]

Combine the real and imaginary parts:

\[ (6 + 3) + (2i - 9i) \]

\[ 9 - 7i \]

So, \((2 - 3i) \times (3 + i) = 9 - 7i\).

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:
1. How do you multiply two complex numbers in general?
2. What is the geometric interpretation of multiplying complex numbers?
3. How do you add and subtract complex numbers?
4. What is the conjugate of a complex number?
5. How do you divide complex numbers?
6. What is the modulus of a complex number?
7. How do you convert a complex number from rectangular form to polar form?
8. How do you find the roots of a complex number?

**Tip:** Remember that \(i^2 = -1\) is fundamental in simplifying expressions involving complex numbers.

Learn how to multiply complex numbers (2-3i)*(3+i) with a detailed step-by-step solution using the distributive property. This problem covers key concepts in complex numbers and includes the fundamental theorem of algebra. Suitable for students in Grades 10-12.

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