Let's solve the expression step by step:

Given expression: $(5-2)^3 \times 2 + \left[-4 + (-7)\right] \div (-2+4)^2$

Step 1: Simplify the parentheses and exponents

1. $5 - 2 = 3$
2. $3^3 = 27$
3. $-2 + 4 = 2$
4. $2^2 = 4$

Step 2: Substitute back and simplify further

Now substitute these results back into the expression: $27 \times 2 + \left[-4 + (-7)\right] \div 4$

Step 3: Simplify the terms inside the brackets

$-4 + (-7) = -11$

Now the expression is: $27 \times 2 + \left(-11\right) \div 4$

Step 4: Perform the multiplication

$54 + \left(-11 \div 4\right)$

Step 5: Simplify the division

$-11 \div 4 = -2.75$

Step 6: Final addition

$54 + (-2.75) = 54 - 2.75 = 51.25$

Final Answer

$\boxed{51.25}$

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How does order of operations change when dealing with division and multiplication?
2. What strategies can be used to simplify complex expressions?
3. How do you handle negative numbers in addition and subtraction?
4. What are some methods to verify your solution?
5. How can exponents affect the overall value of an expression?

Tip: When simplifying expressions, it's helpful to write out each step to avoid mistakes, especially when dealing with negative numbers and fractions.