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Let's solve the two parts step by step, expressing the results in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

### Part (a): 
\[
- \frac{1}{3} i \cdot (9 - 15i)
\]
We will distribute the \(-\frac{1}{3} i\) to both terms inside the parentheses.

1. First term: 
\[
-\frac{1}{3} i \cdot 9 = -\frac{9}{3}i = -3i
\]

2. Second term: 
\[
-\frac{1}{3} i \cdot (-15i) = \frac{15}{3} i^2 = 5i^2
\]
Since \(i^2 = -1\):
\[
5i^2 = 5 \times (-1) = -5
\]

Now, combining the results:
\[
-3i - 5 = -5 - 3i
\]

Thus, the result for part (a) is:
\[
-5 - 3i
\]

### Part (b):
\[
(-5 + 4i)(3 - i)
\]
We will expand using the distributive property (FOIL method):

1. First term: 
\[
-5 \cdot 3 = -15
\]

2. Outer term: 
\[
-5 \cdot (-i) = 5i
\]

3. Inner term: 
\[
4i \cdot 3 = 12i
\]

4. Last term: 
\[
4i \cdot (-i) = -4i^2 = 4 \quad \text{(since } i^2 = -1\text{)}
\]

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

Thus, the result for part (b) is:
\[
-11 + 17i
\]

---

Would you like more details on any of these steps or have further questions?

### Related Questions:
1. How do we handle complex number multiplication in general?
2. Why is \(i^2 = -1\) so important in complex number operations?
3. How would the result differ if you reversed the factors in part (b)?
4. What is the geometric interpretation of multiplying two complex numbers?
5. How can we divide two complex numbers, and what would be the result of \(\frac{-5+4i}{3-i}\)?

### Tip:
Always remember that \(i^2 = -1\), which transforms the imaginary part into real components during multiplication of complex numbers!

Multiply and write the result in the form a + bi.
(a) \(-\frac{1}{3}i(9 - 15i)\)
(b) \((-5 + 4i)(3 - i)\)

Learn how to multiply complex numbers and express the result in the form a + bi. This problem covers the distributive property and the FOIL method in complex number operations, including the key concept that i^2 = -1. Suitable for high school students.

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