$ \frac{(469 + 174)^2 - (469 - 174)^2}{469 \times 174} = x $

### Concept & Formula This is another classic aptitude exam pattern testing your knowledge of derived algebraic formulas. In this case, it utilizes the subtraction of two expanded binomial squares. The specific derived formula is: $$ (a + b)^2 - (a - b)^2 = 4ab $$ ### Step-by-Step Solution Let's replace the large numbers with variables to expose the underlying structure: * Let $a = 469$ * Let $b = 174$ Substitute these variables into the expression given in the question: $$ \frac{(a + b)^2 - (a - b)^2}{a \times b} $$ Let's prove the numerator's behavior by expanding both terms: $$ \text{Numerator} = (a^2 + b^2 + 2ab) - (a^2 + b^2 - 2ab) $$ Carefully distribute the negative sign into the second set of parentheses: $$ = a^2 + b^2 + 2ab - a^2 - b^2 + 2ab $$ The $a^2$ and $b^2$ terms cancel out, leaving only the $ab$ terms: $$ = 2ab + 2ab = 4ab $$ Now, place this simplified numerator back over the denominator: $$ = \frac{4ab}{ab} $$ The $ab$ terms cancel out completely from the top and the bottom: $$ = 4 $$ As demonstrated, the actual numerical values (469 and 174) play no role in determining the final answer. ### Exam Strategy & Shortcut Much like the previous problem, you should memorize this format: $ \frac{(a+b)^2 - (a-b)^2}{ab} $. Any equation following this exact mathematical structure will **ALWAYS evaluate to 4**. Recognizing this instantly saves you minutes of calculation. ### Common Pitfall If you mistakenly write the denominator as $(a^2 + b^2)$ rather than $ab$, you might confuse this identity with the previous one and answer 2. Pay very close attention to whether the operator connecting the terms is a plus or a minus, and what the denominator consists of. ### Final Answer **Therefore, the correct answer is 4.**