Let's break down and solve each expression step by step, using the given values: $a = 5$, $b = 6$, and $c = -2$.

a) $a^2 + (b - 2c)^2$

1. $a^2 = 5^2 = 25$
2. $b - 2c = 6 - 2(-2) = 6 + 4 = 10$
3. $(b - 2c)^2 = 10^2 = 100$

So the expression becomes: $a^2 + (b - 2c)^2 = 25 + 100 = 125$

b) $(3b)^2$

1. $3b = 3 \times 6 = 18$
2. $(3b)^2 = 18^2 = 324$

So, the value of the expression is: $(3b)^2 = 324$

c) $ab^2$

1. $ab^2 = 5 \times 6^2 = 5 \times 36 = 180$

So, the value of the expression is: $ab^2 = 180$

d) $\frac{a \cdot a \cdot c}{b}$

1. $a \cdot a \cdot c = 5 \times 5 \times (-2) = -50$
2. $\frac{a \cdot a \cdot c}{b} = \frac{-50}{6}$

Simplifying: $\frac{-50}{6} \approx -8.33 \quad \text{(rounded to two decimal places)}$

Final Answers:

a) $125$
b) $324$
c) $180$
d) $-8.33$ (rounded)

Would you like any further details or clarification on these steps?

Here are 5 related questions to explore further:

1. How would the expressions change if $c$ were positive?
2. What is the general form of these expressions with variables?
3. How can you simplify the expressions for larger values of $a$ and $b$?
4. Can you derive the derivative of any of these expressions?
5. What happens when $a = b = c = 0$?

Tip: Always perform operations inside parentheses first when evaluating complex expressions.