$1904 \times 1904 = x$

### Concept & Formula When multiplying a number by itself (finding the square), and the number is close to a round base like 1900, we can use the algebraic expansion formula for the square of a binomial sum. $$(a + b)^2 = a^2 + 2ab + b^2$$ ### Step-by-Step Solution * Express 1904 as the sum of a base number and a smaller number: $$1904 = 1900 + 4$$ * Apply the binomial square formula: $$(1900 + 4)^2 = (1900)^2 + 2(1900)(4) + (4)^2$$ * Calculate each term individually: $$(1900)^2 = 3610000$$ $$2 \times 1900 \times 4 = 15200$$ $$(4)^2 = 16$$ * Sum the terms together: $$3610000 + 15200 + 16 = 3625216$$ ### Exam Strategy & Shortcut Instead of full calculation, use the last two digits approach. The last two digits of the square of $1900 + x$ will be determined by the last two digits of $x^2$. Here, $x = 4$, and $4^2 = 16$. The answer must end in 16. This eliminates options (b) and (d). Next, estimate the size. $1900^2 = 3610000$. The addition of the middle term $2ab$ ($15200$) pushes the answer into the $3625000$ range, easily confirming option (c) over the inflated option (a). ### Common Pitfall Resorting to traditional vertical multiplication for large four-digit numbers takes too long and frequently leads to carrying or addition errors across the multiple rows of zeroes. ### Final Answer **Therefore, the correct answer is 3625216.**