Simplify $ \frac{(893 + 786)^2 - (893 - 786)^2}{893 \times 786} $

### Concept & Formula The expression is a direct application of the algebraic identity that relates the square of a sum and the square of a difference. By factoring it algebraically, the large numerical values become irrelevant: $$ (a + b)^2 - (a - b)^2 = 4ab $$ ### Step-by-Step Solution * Let $a = 893$ and $b = 786$. * Substitute these variables into the given mathematical expression: $$ \frac{(a + b)^2 - (a - b)^2}{ab} $$ * Expand the numerator using standard binomial identities: $$ (a^2 + b^2 + 2ab) - (a^2 + b^2 - 2ab) $$ * Distribute the negative sign and simplify the numerator: $$ a^2 + b^2 + 2ab - a^2 - b^2 + 2ab = 4ab $$ * Substitute the simplified numerator back into the fraction: $$ \frac{4ab}{ab} $$ * Cancel the common $ab$ terms from the numerator and denominator to get the final result: $$ 4 $$ ### Exam Strategy & Shortcut Recognize the $\frac{(a+b)^2 - (a-b)^2}{ab}$ pattern immediately. It will ALWAYS evaluate to $4$ because the variables completely cancel out. You can solve this question in one second without even reading the specific numbers provided in the numerator and denominator. ### Common Pitfall A common mistake is attempting to perform the actual arithmetic—adding and subtracting the numbers inside the parentheses, squaring the massive results, and then dividing. This wastes immense time and introduces a very high risk of basic calculation errors. ### Final Answer Therefore, the correct answer is **4**.