Let's solve each part of the problem step by step.

Given:

• $z_1 = -6 - 6i$
• $z_2 = 5 - 6i$

Part (a): $6 \cdot z_1 + 3 \cdot z_2$

First, calculate each term: $6 \cdot z_1 = 6 \cdot (-6 - 6i) = -36 - 36i$ $3 \cdot z_2 = 3 \cdot (5 - 6i) = 15 - 18i$

Now, add them together: $6 \cdot z_1 + 3 \cdot z_2 = (-36 - 36i) + (15 - 18i) = -21 - 54i$

So, the answer to part (a) is: $\boxed{-21 - 54i}$

Part (b): $4 \cdot z_1 \cdot z_2$

First, find the product $z_1 \cdot z_2$: $z_1 \cdot z_2 = (-6 - 6i) \cdot (5 - 6i)$

Using the distributive property (FOIL): $z_1 \cdot z_2 = (-6 \cdot 5) + (-6 \cdot -6i) + (-6i \cdot 5) + (-6i \cdot -6i)$ $z_1 \cdot z_2 = -30 + 36i - 30i + 36i^2$

Since $i^2 = -1$: $z_1 \cdot z_2 = -30 + 6i + 36(-1) = -30 + 6i - 36 = -66 + 6i$

Now, multiply this by 4: $4 \cdot z_1 \cdot z_2 = 4 \cdot (-66 + 6i) = -264 + 24i$

So, the answer to part (b) is: $\boxed{-264 + 24i}$

Part (c): $\frac{z_1}{z_2}$

To find the quotient $\frac{z_1}{z_2}$, multiply the numerator and denominator by the conjugate of the denominator:

Let $z_2^* = 5 + 6i$ (the conjugate of $z_2$).

$\frac{z_1}{z_2} = \frac{-6 - 6i}{5 - 6i} \cdot \frac{5 + 6i}{5 + 6i} = \frac{(-6 - 6i)(5 + 6i)}{(5 - 6i)(5 + 6i)}$

First, calculate the numerator: $(-6 - 6i)(5 + 6i) = -30 - 36i - 30i - 36i^2 = -30 - 66i + 36 = 6 - 66i$

Next, calculate the denominator: $(5 - 6i)(5 + 6i) = 25 + 30i - 30i - 36i^2 = 25 + 36 = 61$

Thus: $\frac{z_1}{z_2} = \frac{6 - 66i}{61} = \frac{6}{61} - \frac{66i}{61}$

So, the answer to part (c) is: $\boxed{\frac{6}{61} - \frac{66i}{61}}$

Summary of Answers:

a) $-21 - 54i$
b) $-264 + 24i$
c) $\frac{6}{61} - \frac{66i}{61}$

Would you like more details or have any other questions?

Here are 5 related questions:

1. How do you multiply two complex numbers?
2. What is the geometric interpretation of multiplying complex numbers?
3. How do you find the conjugate of a complex number?
4. Can you explain the process of dividing complex numbers using conjugates?
5. What are the properties of the magnitude of a complex number?

Tip: When multiplying complex numbers, remember to treat $i^2$ as $-1$ to simplify your expressions.