Let's simplify the given expression step-by-step:

The expression is:

$4 - 5(-2)(1) + 1$

1. First, handle the multiplication inside the expression:

   $-5 \times -2 \times 1 = 10$

2. Substitute this back into the expression:

   $4 + 10 + 1$

3. Finally, add the terms together:

   $4 + 10 + 1 = 15$

So, the simplified result is:

$\boxed{15}$

Would you like more details on the steps? Here are some related questions:

1. How do you handle multiple operations in an expression?
2. What is the order of operations, and why is it important?
3. How do negative signs affect multiplication?
4. Can parentheses change the outcome of an expression?
5. What are some common mistakes when simplifying expressions?

Tip: Always handle multiplication and division before addition and subtraction to avoid mistakes with order of operations.