The problem shown in the image is:

Simplify:

$\left( \left( 993 \right)^2 - \left( 992 \right)^2 \right) + 5 \times 47 + \left( \left( 93 \right)^2 - \left( 92 \right)^2 \right)$

Solution:

1. Use the difference of squares formula for the terms of the form $a^2 - b^2$, which is:

$a^2 - b^2 = (a - b)(a + b)$

• For $\left( 993^2 - 992^2 \right)$, we have $a = 993$ and $b = 992$. So:

$993^2 - 992^2 = (993 - 992)(993 + 992) = 1 \times 1985 = 1985$

• For $\left( 93^2 - 92^2 \right)$, we have $a = 93$ and $b = 92$. So:

$93^2 - 92^2 = (93 - 92)(93 + 92) = 1 \times 185 = 185$

2. Now substitute these values back into the original expression:

$1985 + 5 \times 47 + 185$

3. Calculate $5 \times 47$:

$5 \times 47 = 235$

4. Add the results:

$1985 + 235 + 185 = 2405$

Final Answer:

The simplified value of the expression is 2405.

Would you like more details or have any questions?
Here are 5 related questions for further practice:

1. Can you simplify another expression involving the difference of squares, such as $(105^2 - 104^2) + (54^2 - 53^2)$?
2. What is the general formula for the difference of cubes and how does it compare to the difference of squares?
3. If $(x^2 - y^2)$ is factored, how would you solve it for various values of $x$ and $y$?
4. How does this technique help in simplifying larger polynomials?
5. Can you think of a real-life scenario where the difference of squares might be useful?

Tip: The difference of squares can quickly simplify large numbers when the values are close to each other, saving time on direct squaring.