3[(-3)^2 -2(4)^2 -(-21)] -6(-5) / -3^2 -(-3)(-1)

Let's solve the expression step by step, following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)):

Given expression:  
\[ 3\left[(-3)^2 - 2(4)^2 - (-21)\right] - \frac{6(-5)}{-3^2} - (-3)(-1) \]

### Step 1: Evaluate the exponents
- \((-3)^2 = 9\)
- \((4)^2 = 16\)
- \(-3^2\) (interpreted as \(-(3^2)\)) = \(-9\)

So, the expression becomes:
\[ 3\left[9 - 2(16) - (-21)\right] - \frac{6(-5)}{-9} - (-3)(-1) \]

### Step 2: Multiply within the brackets and simplify
- \(2(16) = 32\)
- So the expression inside the brackets becomes:
  \[ 9 - 32 - (-21) = 9 - 32 + 21 = -23 + 21 = -2 \]

So, now the expression is:
\[ 3(-2) - \frac{6(-5)}{-9} - (-3)(-1) \]

### Step 3: Multiply and simplify the remaining terms
- \(3(-2) = -6\)
- \(6(-5) = -30\)
- So, the next term becomes:
  \[ \frac{-30}{-9} = \frac{30}{9} = \frac{10}{3} \]
- \((-3)(-1) = 3\)

Now the expression is:
\[ -6 + \frac{10}{3} - 3 \]

### Step 4: Combine the terms
First, combine the integers:
\[ -6 - 3 = -9 \]

Now, add \(\frac{10}{3}\) to \(-9\) (convert \(-9\) to a fraction for easier addition):
\[ -9 = \frac{-27}{3} \]
\[ \frac{-27}{3} + \frac{10}{3} = \frac{-27 + 10}{3} = \frac{-17}{3} \]

### Final Answer
\[ \frac{-17}{3} \]

The result of the expression is \(\frac{-17}{3}\).

Would you like further details or explanations?

Here are five related questions to consider:
1. How does the solution change if the base of \((-3)^2\) is positive instead of negative?
2. What happens if you change the exponent of \(-3\) in \(-3^2\) to 3?
3. Can you evaluate \(4\left[(2)^2 - 3(3)^2 - (-12)\right] - 5(-6) / -2^2 - (-2)(1)\)?
4. How would the expression change if the denominator in \(\frac{6(-5)}{-9}\) were positive?
5. What is the impact of changing the multiplication in \((-3)(-1)\) to division?

**Tip:** When dealing with expressions that include negative numbers raised to powers, be mindful of whether the negative sign is included inside or outside the parentheses, as it affects the result.

Learn how to solve a complex mathematical expression step-by-step. This problem involves evaluating exponents, performing multiplication and division operations, and applying the order of operations (PEMDAS). Suitable for students in Grades 7-9.

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