Let's evaluate the given expression step-by-step, following the order of operations (PEMDAS/BODMAS):

The expression is:
$\left[ 5 + (-4)^2 \cdot 2 \right] + 4^2$

Step 1: Handle exponents

1. $(-4)^2 = 16$
2. $4^2 = 16$

Now the expression becomes:
$\left[ 5 + 16 \cdot 2 \right] + 16$

Step 2: Perform multiplication inside the brackets

$16 \cdot 2 = 32$

Now the expression becomes:
$\left[ 5 + 32 \right] + 16$

Step 3: Perform addition inside the brackets

$5 + 32 = 37$

Now the expression becomes:
$37 + 16$

Step 4: Perform the final addition

$37 + 16 = 53$

Final Answer:

$53$

Would you like me to explain any part of the solution further?

Here are 5 related questions to practice:

1. How would the result change if $-4$ were not squared?
2. What is the role of parentheses in ensuring the correct order of operations?
3. How do you evaluate expressions with multiple exponents?
4. What happens if multiplication and addition are swapped in priority by mistake?
5. Can you create a similar problem but with three layers of brackets?

Tip: Always simplify expressions step-by-step, starting with exponents and working your way through multiplication/division before addition/subtraction.