Evaluate the expression 217 * 217 + 183 * 183 using correct arithmetic. Choose the correct value of the final result.

Introduction:
This problem tests quick simplification and accurate squaring. Expressions like a*a + b*b are easiest when you compute each square carefully using (u + v)^2 = u^2 + 2uv + v^2. That reduces mental mistakes compared to long multiplication. After finding both squares, add them to get the final value. Accuracy matters because the options are close-looking numbers.

Given Data / Assumptions:

• Compute 217 * 217
• Compute 183 * 183
• Add the two results
• Use (u + v)^2 expansion to simplify squaring

Concept / Approach:
Write 217 as (200 + 17) and 183 as (180 + 3). Then square each using (u + v)^2 = u^2 + 2uv + v^2. Finally, add the two square values to obtain the required sum.

Step-by-Step Solution:
217^2 = (200 + 17)^2 = 200^2 + 2*200*17 + 17^2 = 40000 + 6800 + 289 = 47089 183^2 = (180 + 3)^2 = 180^2 + 2*180*3 + 3^2 = 32400 + 1080 + 9 = 33489 Required sum = 47089 + 33489 = 80578

Verification / Alternative check:
A quick reasonableness check: 217^2 is a bit less than 220^2 = 48400 and 183^2 is a bit more than 180^2 = 32400, so total near 80800. The exact 80578 fits this range well.

Why Other Options Are Wrong:
They come from common arithmetic slips: missing the 2uv term, mis-adding the final sum, or squaring 217/183 incorrectly by a few hundreds or thousands.

Common Pitfalls:
Using (u+v)^2 = u^2 + v^2 (forgetting 2uv), or making an addition error when combining the two squared results.

Final Answer:
The value of 217 * 217 + 183 * 183 is 80,578.