https://file.aiquickdraw.com/mathbot/file/1729195206297be2pv671.jpg

Let's solve the given problems from the image.

### Problem (a)
\[
-\frac{1}{3} i (9 - 15i)
\]
We need to distribute the \(-\frac{1}{3} i\) across the expression \(9 - 15i\).

1. First, multiply \(-\frac{1}{3} i\) with 9:
   \[
   -\frac{1}{3} i \times 9 = -3i
   \]

2. Then, multiply \(-\frac{1}{3} i\) with \(-15i\):
   \[
   -\frac{1}{3} i \times -15i = \frac{15}{3} i^2 = 5 \times (-1) = -5
   \]
   (Remember, \(i^2 = -1\)).

So, the result is:
\[
-3i - 5
\]
Thus, in the form \(a + bi\), the result is:
\[
-5 - 3i
\]

### Problem (b)
\[
(-5 + 4i)(3 - i)
\]
We need to expand the product using the distributive property (FOIL method):

1. Multiply \(-5\) with 3:
   \[
   -5 \times 3 = -15
   \]

2. Multiply \(-5\) with \(-i\):
   \[
   -5 \times (-i) = 5i
   \]

3. Multiply \(4i\) with 3:
   \[
   4i \times 3 = 12i
   \]

4. Multiply \(4i\) with \(-i\):
   \[
   4i \times (-i) = -4i^2 = -4 \times (-1) = 4
   \]
   (Again, using \(i^2 = -1\)).

Now, combine all the terms:
\[
-15 + 5i + 12i + 4 = -15 + 4 + (5i + 12i) = -11 + 17i
\]

Thus, the result in the form \(a + bi\) is:
\[
-11 + 17i
\]

---

Would you like further clarification on any step? Here are 5 related questions to expand your understanding:

1. What is the significance of \(i^2 = -1\) in complex number multiplication?
2. How would you multiply two general complex numbers, say \((a + bi)(c + di)\)?
3. Why do we always express complex numbers in the form \(a + bi\)?
4. What is the geometric interpretation of multiplying two complex numbers?
5. How do you find the conjugate of a complex number, and why is it useful?

**Tip:** When multiplying complex numbers, always remember to replace \(i^2\) with \(-1\).

Multiply and write the result in the form a + bi: 
a. (-1/3) * i * (9 - 15i)
b. (-5 + 4i)(3 - i)

Learn how to multiply complex numbers such as (-1/3)i(9 - 15i) and (-5 + 4i)(3 - i) using the distributive property and FOIL method. This step-by-step solution covers key concepts like the imaginary unit (i^2 = -1) and complex number multiplication. Suitable for students in Grades 10-12.

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