$1256 \times 3892 = x$

Concept & Strategy This is a straight arithmetic question requiring the multiplication of two four-digit numbers. Breaking down one of the numbers into simpler components (distributive property) is the most reliable manual method. Step-by-Step Solution * We need to calculate: $$1256 \times 3892$$ * Let's break down $3892$ into $(4000 - 100 - 8)$ for easier multiplication. $$1256 \times (4000 - 100 - 8)$$ * Multiply $1256$ by each component: $$1256 \times 4000 = 5024000$$ $$1256 \times 100 = 125600$$ $$1256 \times 8 = 10048$$ * Now combine them according to the signs: $$5024000 - 125600 - 10048$$ * Subtract $125600$ from $5024000$: $$5024000 - 125600 = 4898400$$ * Finally, subtract $10048$: $$4898400 - 10048 = 4888352$$ Exam Strategy & Shortcut Let's use the last two digits to eliminate options. Multiply the last two digits of both numbers: $56 \times 92$. $56 \times (100 - 8) = 5600 - 448 = 5152$. The product must end in $52$. Options (a), (c), and (e) are left. Now use the digital root (sum of digits until a single digit is reached): Digital root of $1256 = 1+2+5+6 = 14 \rightarrow 1+4 = 5$. Digital root of $3892 = 3+8+9+2 = 22 \rightarrow 2+2 = 4$. Product of digital roots = $5 \times 4 = 20 \rightarrow 2+0 = 2$. Check digital roots of remaining options: (a) $4883852 \rightarrow 4+8+8+3+8+5+2 = 38 \rightarrow 11 \rightarrow 2$. (Possible) (c) $4888352 \rightarrow 4+8+8+8+3+5+2 = 38 \rightarrow 11 \rightarrow 2$. (Possible) We need to evaluate deeper. If we approximate: $1250 \times 3900 = 4875000$. Our exact number should be slightly higher. $4888352$ is a better fit than $4883852$. Common Pitfall Doing a standard 4-by-4 long multiplication is extremely prone to addition errors in the middle rows. Breaking it into manageable chunks reduces this risk. Final Answer **Therefore, the correct answer is 4888352.**