$397 \times 397 + 104 \times 104 + 2 \times 397 \times 104 = x$

### Concept & Formula This expression exactly matches the expanded form of a perfect square trinomial. Recognizing this pattern eliminates the need for any manual multiplication. $$a^2 + b^2 + 2ab = (a + b)^2$$ ### Step-by-Step Solution * Identify the repeating terms in the given expression and assign them to variables: * Let $a = 397$ * Let $b = 104$ * Rewrite the numerical expression using these assigned variables: $$a \times a + b \times b + 2 \times a \times b$$ $$a^2 + b^2 + 2ab$$ * Apply the perfect square trinomial identity to condense the expression: $$(a + b)^2$$ * Substitute the original numerical values back into the condensed formula: $$(397 + 104)^2$$ * Perform the basic addition inside the parentheses first: $$(501)^2$$ * Square the resulting number rapidly by splitting it into $(500 + 1)^2$: $$500^2 + 1^2 + 2(500)(1) = 250000 + 1 + 1000 = 251001$$ ### Exam Strategy & Shortcut Use the **Unit Digit Method** to cross-check. The expression condenses to $(397 + 104)^2 = (501)^2$. The unit digit of 501 is 1, so the square must end in 1. All options end in 1, so unit digit alone isn't enough. However, $500^2$ is exactly 250,000. Because 501 is slightly larger than 500, the answer must be slightly larger than 250,000. Option (b), 251,001, fits perfectly without doing the full expansion. ### Common Pitfall Attempting to multiply $397 \times 397$ and $104 \times 104$ manually is a massive trap. It consumes excessive time and highly increases the likelihood of arithmetic errors. Always scan for algebraic patterns before calculating. ### Final Answer Therefore, the correct answer is **251001**.