$$\frac{(489 + 375)^2 - (489 - 375)^2}{(489 \times 375)} = x$$

### Concept & Formula This fraction follows a universal algebraic template. The difference between the square of a sum and the square of a difference always simplifies to exactly four times the product of the two numbers. $$(a + b)^2 - (a - b)^2 = 4ab$$ ### Step-by-Step Solution * Analyze the structure of the numerator and assign variables to the recurring numbers to reveal the underlying algebra: * Let $a = 489$ * Let $b = 375$ * Substitute these variables back into the entire original fractional expression: $$\frac{(a + b)^2 - (a - b)^2}{ab}$$ * Apply the standard algebraic identity to simplify the numerator: $$(a + b)^2 - (a - b)^2 = 4ab$$ * Replace the numerator in our fraction with this highly simplified form: $$\frac{4ab}{ab}$$ * Cancel out the common $ab$ terms from both the numerator and the denominator: $$4$$ * Because the variables completely cancel out, the actual numerical values of $a$ (489) and $b$ (375) are entirely irrelevant to the final outcome. ### Exam Strategy & Shortcut Memorize the identity $\frac{(a+b)^2 - (a-b)^2}{ab} = 4$. Whenever you see this exact structural format in a competitive exam, immediately mark 4 as the answer. Do not write anything down or look at the actual numbers provided, as they are merely distractors. ### Common Pitfall A critical mistake is attempting to add and subtract the numbers inside the parentheses, squaring the massive results, and then trying to perform long division. This will drain minutes of exam time and almost guarantee an arithmetic error. ### Final Answer Therefore, the correct answer is **4**.