The problem asks you to simplify the expression:

$((993)^2 - (992)^2) + 5 \times 47 + ((193)^2 - (192)^2)$

Step-by-Step Solution:

1. Use the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$

   • For $(993)^2 - (992)^2$: $993^2 - 992^2 = (993 - 992)(993 + 992) = 1 \times 1985 = 1985$
   • For $(193)^2 - (192)^2$: $193^2 - 192^2 = (193 - 192)(193 + 192) = 1 \times 385 = 385$

2. Simplify the original expression: $1985 + 5 \times 47 + 385$

3. Calculate $5 \times 47$: $5 \times 47 = 235$

4. Add the results: $1985 + 235 + 385 = 2605$

Final Answer:

The simplified result of the expression is 2605.

Would you like further details or have any questions?

Follow-up Questions:

1. What is the difference of squares formula and how is it applied?
2. Can you solve another similar expression with different numbers?
3. How does changing the values in the expression affect the final result?
4. What would happen if you added or subtracted a constant to the entire expression?
5. How would the solution change if the multiplication was a different operation?

Tip:

Always look for patterns or identities like the difference of squares to simplify expressions more easily.