Let's walk through the expression step by step.

The given expression is:

$\frac{1}{2} \left[ \frac{3}{4} (5 - 9)^2 + \frac{1}{4} (8 - 12)^2 \right]$

Step 1: Simplify the expressions inside the parentheses.

• $5 - 9 = -4$
• $8 - 12 = -4$

Step 2: Square the results.

• $(-4)^2 = 16$
• $(-4)^2 = 16$

Now the expression becomes:

$\frac{1}{2} \left[ \frac{3}{4} \cdot 16 + \frac{1}{4} \cdot 16 \right]$

Step 3: Perform the multiplications.

• $\frac{3}{4} \cdot 16 = 12$
• $\frac{1}{4} \cdot 16 = 4$

Now the expression becomes:

$\frac{1}{2} \left[ 12 + 4 \right]$

Step 4: Add the terms inside the brackets.

$12 + 4 = 16$

Now the expression becomes:

$\frac{1}{2} \cdot 16$

Step 5: Perform the final multiplication.

$\frac{1}{2} \cdot 16 = 8$

Final Answer:

$\boxed{8}$

Would you like any further details or clarifications?

Here are 5 related questions for deeper practice:

1. How do you distribute fractions when simplifying similar expressions?
2. What are the rules for squaring negative numbers?
3. How would the answer change if one of the squared terms was $(-4)^3$ instead of $(-4)^2$?
4. Can you simplify the expression using factorization methods instead?
5. How would the result change if we had to multiply by $\frac{3}{2}$ instead of $\frac{1}{2}$?

Tip: Always perform operations inside parentheses first according to the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).