$287 \times 287 + 269 \times 269 - 2 \times 287 \times 269 = x$

### Concept & Formula The given expression strictly follows a well-known algebraic identity for the square of a binomial difference. Recognizing this pattern collapses a complex arithmetic problem into a simple mental math exercise. $$a^2 + b^2 - 2ab = (a - b)^2$$ ### Step-by-Step Solution * Map the expression to the algebraic identity variables: Let $a = 287$ and $b = 269$. * Rewrite the original problem using these variables: $$(287 \times 287) + (269 \times 269) - (2 \times 287 \times 269) \rightarrow a^2 + b^2 - 2ab$$ * Collapse the expression into its factored form: $$(287 - 269)^2$$ * Calculate the difference inside the parentheses: $$287 - 269 = 18$$ * Square the result to find the final value: $$(18)^2 = 324$$ ### Exam Strategy & Shortcut You should have the squares of all numbers up to 30 memorized for competitive exams. Once you spot the $a^2 + b^2 - 2ab$ pattern, instantly subtract the two unique numbers ($287 - 269 = 18$). Then, either recall that $18^2 = 324$, or use unit digits: the unit digit of $(18)^2$ is the unit digit of $8 \times 8 = 64$, which is 4. Option (d) 324 and (a) 534 end in 4, but $18^2$ is much smaller than 500, leading straight to 324. ### Common Pitfall Brute-forcing the multiplication. Even partial brute-forcing (like only calculating the unit digit of the entire expanded expression) takes longer and is more error-prone than applying the identity. ### Final Answer **Therefore, the correct answer is 324.**