Let's break down and solve the expression step by step:

### Given:
\[ 4^2 + 3|4x - 9| \]
with \( x = -2 \).

### Step 1: Simplify the Powers
First, simplify \( 4^2 \):
\[
4^2 = 16
\]
Now the expression becomes:
\[
16 + 3|4x - 9|
\]

### Step 2: Substitute \( x = -2 \) into the Absolute Value Expression
Substitute \( x = -2 \) into \( 4x - 9 \):
\[
4(-2) - 9 = -8 - 9 = -17
\]
Now we have:
\[
16 + 3| -17 |
\]
Since the absolute value of \( -17 \) is \( 17 \), this simplifies to:
\[
16 + 3(17)
\]

### Step 3: Simplify the Multiplication
Now, multiply:
\[
3(17) = 51
\]

### Step 4: Add the Terms
Finally, add:
\[
16 + 51 = 67
\]

Thus, the value of the expression is:
\[
\boxed{67}
\]

Would you like more details or have any questions about this solution?

### Related Questions:
1. What is the general method to simplify expressions involving absolute values?
2. How does the substitution of variables affect the structure of the expression?
3. Can you explain how absolute values change the output of negative inputs?
4. How does exponentiation affect the result when simplifying an expression?
5. What happens if you change \( x \) to a different value, say \( x = 3 \), in this problem?

### Tip:
Always handle the absolute value separately before continuing with any other arithmetic to avoid errors in your calculations.

Learn how to solve the expression 4^2 + 3|4x - 9| by substituting x = -2 and simplifying step-by-step. This problem covers exponents, absolute value, and substitution concepts, suitable for Grades 8-10 students.

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