Number System — Evaluate: 387 × 387 + 114 × 114 + 2 × 387 × 114. Identify the exact value.

Introduction / Context:
Recognize the classic binomial identity embedded in the expression. The sum of two squares plus twice their product equals the square of the sum. Spotting this identity saves time and eliminates arithmetic burden.

Given Data / Assumptions:

• Expression: 387^2 + 114^2 + 2 × 387 × 114.
• Numbers are integers; exact evaluation is required.
• Binomial identity applies directly.

Concept / Approach:
Use (x + y)^2 = x^2 + y^2 + 2xy. Here, x = 387 and y = 114. Therefore, the entire expression collapses neatly to (387 + 114)^2, which is much faster to compute accurately.

Step-by-Step Solution:
1) Identify x = 387, y = 114.2) Compute x + y = 387 + 114 = 501.3) Evaluate (x + y)^2 = 501^2.4) 501^2 = (500 + 1)^2 = 500^2 + 2*500*1 + 1^2 = 250000 + 1000 + 1 = 251001.

Verification / Alternative check:
Direct calculation of the original expression using the identity steps confirms the same result. The near-500 square check (500^2 = 250000) ensures the magnitude is sensible, with the extra 1001 added giving 251001.

Why Other Options Are Wrong:
250001 omits the 2xy and most of the linear term; 260101 and 261001 overshoot the true square; 251121 is a common mis-square of 501 (it would correspond to 511^2 ≈ 261121, not applicable).

Common Pitfalls:
Forgetting the 2xy term; computing 501^2 as 500^2 + 1^2 only; arithmetic slips when adding 1000 + 1 to 250000.

Final Answer:
251001