Let's simplify the expression step-by-step:

The expression is:

\[
4(3)^2 - 3(3)(4) + 2(4)^2
\]

1. Calculate the squares:
   - \( (3)^2 = 9 \)
   - \( (4)^2 = 16 \)

2. Substitute these values back into the expression:

\[
4 \cdot 9 - 3 \cdot 3 \cdot 4 + 2 \cdot 16
\]

3. Perform the multiplications:
   - \( 4 \cdot 9 = 36 \)
   - \( 3 \cdot 3 \cdot 4 = 36 \)
   - \( 2 \cdot 16 = 32 \)

4. Substitute these results back:

\[
36 - 36 + 32
\]

5. Simplify the expression:

\[
0 + 32 = 32
\]

### Final Answer:

\[
\boxed{32}
\]

Would you like more details on the steps? Here are some related questions:

1. How do you handle similar expressions with more variables?
2. What is the significance of the order of operations in simplifying expressions?
3. Can you explain how to factor expressions like this one?
4. How would this expression change if different exponents were used?
5. How can we verify our results for expressions like these?

**Tip:** Always follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) when simplifying mathematical expressions.

Master the process of simplifying the algebraic expression 4(3)^2 - 3(3)(4) + 2(4)^2 with step-by-step guidance. This example covers essential algebraic concepts, including the order of operations (PEMDAS) and simplification techniques, making it ideal for students in Grades 6-8.

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