The problem states:

If $a \# b = 2a - b$ and $c * d = c \cdot d^2$, evaluate $(-4) \# (3 * 2)$.

Let's break it down step by step:

1. Evaluate the expression $3 * 2$:

   • From the definition, $c * d = c \cdot d^2$. Substituting $c = 3$ and $d = 2$: $3 * 2 = 3 \cdot 2^2 = 3 \cdot 4 = 12$

2. Now evaluate $(-4) \# 12$:

   • From the definition, $a \# b = 2a - b$. Substituting $a = -4$ and $b = 12$: $(-4) \# 12 = 2(-4) - 12 = -8 - 12 = -20$

Thus, the final answer is $\boxed{-20}$.

Would you like more details on any part of the solution? Here are five related problems to practice:

1. What is $5 \# (4 * 3)$ given the same definitions?
2. Can you simplify $2 \# (5 * 1)$?
3. How would the evaluation change if $a \# b = a + 2b$ instead?
4. What would be $(-6) \# (2 * 5)$ based on the same function?
5. How does the order of operations affect the evaluation of similar expressions?

Tip: When dealing with custom operations like $\#$ or $*$, carefully follow the given definitions to avoid confusion.