Simplify [(25)⁴ + 1250 + 1 − (25)²] ⁄ [(25)² + 25 + 1].

Correct — C. Put a = 25 and rewrite 1250 as 2a², since 2 × 625 = 1250.

Numerator = a⁴ + 2a² + 1 − a² = a⁴ + a² + 1

a⁴ + a² + 1 = (a² + 1)² − a² = (a² + a + 1)(a² − a + 1)
The denominator is a² + a + 1, so it cancels.

Value = a² − a + 1 = 625 − 25 + 1 = 601 → option (c)

Why the others are wrong

• (a)
  604 — 604 fails the direct division. The denominator is 651 (625 + 25 + 1) and 651 × 604 = 393,204, against a numerator of 391,251.
• (b)
  602 — 602 overshoots by exactly one denominator: 651 × 602 = 391,902, which is 651 more than the numerator 391,251.
• (d)
  603 — 603 overshoots by two denominators: 651 × 603 = 392,553, or 1,302 above the numerator 391,251.

The identity worth holding is a⁴ + a² + 1 = (a² + a + 1)(a² − a + 1). It comes from adding and subtracting a²: a⁴ + 2a² + 1 − a² is (a² + 1)² − a², a difference of squares.

The question hides the identity by printing 2a² as the bare number 1250. Spotting that 1250 = 2 × 25² is the whole difficulty.

So when a denominator reads a² + a + 1, look upstairs for a⁴ + a² + 1. Where the numerator is exactly that, the quotient is a² − a + 1 and no long division is needed.

Key facts

• a⁴ + a² + 1 = (a² + 1)² − a² = (a² + a + 1)(a² − a + 1).
• With a = 25, the printed 1250 is 2a², which is what turns the numerator into a⁴ + a² + 1.
• The numerator evaluates to 391,251 and the denominator to 651.
• 391,251 ÷ 651 = 601 exactly, matching a² − a + 1 = 625 − 25 + 1.

Common traps

• Treating 1250 as an unrelated constant instead of 2 × 25².
• Reading the printed 1250 as a² rather than 2a², which leaves a⁴ + 1 upstairs and nothing for the denominator to cancel.
• Slipping a single unit in 625 − 25 + 1, which every option here is designed to catch.

No directly related past PYQ was found.