Let's simplify the expression step by step using the correct order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

The expression is:

$\left( 12 - \frac{3}{5} \times \sqrt{25} \right)^2 + (2 - 4)^2$

Step 1: Simplify the square roots

We know that $\sqrt{25} = 5$. Substituting this value in, the expression becomes:

$\left( 12 - \frac{3}{5} \times 5 \right)^2 + (2 - 4)^2$

Step 2: Simplify the multiplication

$\frac{3}{5} \times 5 = 3$ Now the expression is:

$\left( 12 - 3 \right)^2 + (2 - 4)^2$

Step 3: Perform the subtraction inside the parentheses

$12 - 3 = 9$ $2 - 4 = -2$ Now the expression is:

$(9)^2 + (-2)^2$

Step 4: Apply the exponents

$9^2 = 81$ $(-2)^2 = 4$ Now the expression is:

$81 + 4$

Step 5: Final addition

$81 + 4 = 85$

The simplified result is 85.

Correct Answer:

c) 85

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Would you like further clarification or details on any step? Here are some related questions to explore:

1. How do radicals affect the order of operations in more complex expressions?
2. Can a square root ever result in a negative number?
3. What happens if you multiply different types of radicals?
4. How would the process change with cube roots or other higher roots?
5. Why does squaring a negative number result in a positive value?

Tip: Always simplify expressions inside parentheses first, especially when dealing with radicals or exponents.