Let's break down the expression step by step:

The given expression is:

$7 + \left[ 2 + 3 \times (-5) \right]$

Step 1: Handle the multiplication first (following the order of operations, PEMDAS/BODMAS).

$3 \times (-5) = -15$

Now, the expression becomes: $7 + \left[ 2 + (-15) \right]$

Step 2: Simplify inside the brackets.

$2 + (-15) = -13$

Now, the expression is: $7 + (-13)$

Step 3: Simplify the final addition.

$7 + (-13) = -6$

Thus, the value of the expression is -6.

Let me know if you'd like more details or have any questions!

Here are some related questions to explore:

1. What is the order of operations (PEMDAS/BODMAS) and why is it important in simplifying expressions?
2. How do you simplify expressions involving multiple operations like addition, subtraction, multiplication, and division?
3. How can we apply the distributive property to simplify more complex expressions?
4. What happens when you multiply a positive number by a negative number?
5. How do parentheses change the order of operations?

Tip: Always simplify the operations inside parentheses first, then handle multiplication or division before addition and subtraction.