Simplify and evaluate the expression 213 × 213 + 187 × 187 using suitable algebraic identities or direct calculation.

Introduction / Context:
This question checks basic arithmetic with large numbers and recognition of patterns involving squares. The expression 213 × 213 + 187 × 187 is the same as 213^2 + 187^2. While it can be computed directly, one may also look for algebraic shortcuts or numeric patterns. However, even without a special identity, systematic multiplication and addition can quickly yield the exact value.

Given Data / Assumptions:

• Expression: 213 × 213 + 187 × 187.

• All numbers are integers and we seek an exact integer result.

• No approximations or calculators are required; accurate manual computation or structured reasoning is expected.

Concept / Approach:
We can evaluate each square separately and then add the results. For 213^2, write it as (200 + 13)^2 or perform standard column multiplication. The same applies for 187^2. Alternatively, we can use the identity (a + b)^2 = a^2 + 2ab + b^2 to reduce mental workload. After finding both squares, we add them to get the final value and then match it against the options.

Step-by-Step Solution:
Step 1: Compute 213^2. Write 213 as 200 + 13. Step 2: Using the identity (a + b)^2 = a^2 + 2ab + b^2, we get 213^2 = 200^2 + 2 × 200 × 13 + 13^2. Step 3: Calculate these terms: 200^2 = 40000, 2 × 200 × 13 = 5200, and 13^2 = 169. Step 4: Add them: 40000 + 5200 + 169 = 45369. Step 5: Compute 187^2. Write 187 as 200 − 13. Step 6: Apply (a − b)^2 = a^2 − 2ab + b^2: 187^2 = 200^2 − 2 × 200 × 13 + 13^2. Step 7: Calculate: 200^2 = 40000, −2 × 200 × 13 = −5200, and 13^2 = 169. Step 8: Add them: 40000 − 5200 + 169 = 350? Wait carefully: 40000 − 5200 = 34800, plus 169 = 34969. Step 9: Now add the two squares: 213^2 + 187^2 = 45369 + 34969 = 80338.

Verification / Alternative check:
We can check the addition: 45369 + 34969. Add units: 9 + 9 = 18, write 8 carry 1. Tens: 6 + 6 + 1 = 13, write 3 carry 1. Hundreds: 3 + 9 + 1 = 13, write 3 carry 1. Thousands: 5 + 4 + 1 = 10, write 0 carry 1. Ten thousands: 4 + 3 + 1 = 8. The result is 80338, which matches the earlier sum. This confirms that the calculation is accurate.

Why Other Options Are Wrong:
The options 80438 and 81338 are close but differ by 100 or 1000, errors that could come from misplacing carries during addition. The options 60338 and 60438 are much smaller and would require significantly smaller squares, which is inconsistent with 213^2 and 187^2 both being near 40000. Only 80338 matches the exact computed value of the expression.

Common Pitfalls:
A typical mistake is mismanaging carries during multi digit addition or subtraction, especially when mental calculation is attempted too quickly. Some learners may attempt to compute the squares by repeated multiplication without using identities, increasing the chance of error. Using (200 ± 13)^2 reduces mental load and makes each step explicit. Double checking intermediate sums is a good habit for avoiding small arithmetic slips in exam conditions.

Final Answer:
The value of 213 × 213 + 187 × 187 is 80338.