$(80)^2 - (65)^2 + 81 = x$

### Concept & Formula Simplify the squared terms first by applying the difference of two squares identity before adding the final constant. $$a^2 - b^2 = (a - b)(a + b)$$ ### Step-by-Step Solution * First, isolate the squared terms to process them efficiently: $(80)^2 - (65)^2$. * Apply the identity where $a = 80$ and $b = 65$: $$(80 - 65)(80 + 65) = (15)(145)$$ * Calculate the resulting product by splitting the terms logically: $$15 \times 145 = 15 \times (100 + 45) = 1500 + 675 = 2175$$ * Add the remaining constant of 81 to this intermediate result: $$2175 + 81 = 2256$$ ### Exam Strategy & Shortcut You can utilize the **Unit Digit Method** to eliminate options rapidly. The number $(80)^2$ ends in 0, and $(65)^2$ ends in 5. So, their difference ends in $10 - 5 = 5$. Then, add the unit digit of 81 (which is 1). $5 + 1 = 6$. The correct answer must end in 6, leaving only options (a) and (d). Since $80^2 = 6400$ and $65^2 = 4225$, the difference is roughly 2200. Option (d) matches this perfectly. ### Common Pitfall Calculating $65^2$ manually can take unnecessary time if you don't know the standard ending-in-5 shortcut. Utilizing the $a^2 - b^2$ formula avoids large manual squaring equations entirely. ### Final Answer Therefore, the correct answer is **2256**.