Simplify $387 \times 387 + 113 \times 113 + 2 \times 387 \times 113$

### Concept & Formula The expression follows the algebraic identity for the square of a binomial sum. We can avoid massive calculations by substituting the numbers into the pattern: $$ (a + b)^2 = a^2 + b^2 + 2ab $$ ### Step-by-Step Solution * Let $a = 387$ and $b = 113$. * The given expression is $(387)^2 + (113)^2 + 2 \times 387 \times 113$, which matches the exact form of $a^2 + b^2 + 2ab$. * According to the algebraic formula, this expression is equal to $(a + b)^2$. * Substitute the values of $a$ and $b$ back into the simplified formula: $$ (387 + 113)^2 $$ * Add the numbers inside the parenthesis: $$ (500)^2 $$ * Calculate the final square: $$ 250000 $$ ### Exam Strategy & Shortcut Never multiply out 3-digit numbers in this format. The presence of two squared terms and a $2ab$ term is a giant billboard indicating an algebraic identity. Furthermore, note that $387 + 113$ sums perfectly to $500$, a clean, round number. Examiners design these questions specifically to reward pattern recognition over brute-force arithmetic. ### Common Pitfall A common mistake is failing to recognize the pattern and attempting to calculate $387 \times 387$ manually. This wastes valuable exam time and heavily increases the chance of a simple multiplication error costing you the mark. ### Final Answer Therefore, the correct answer is **250000**.