Ifx + 1⁄x = −1, then compute: x^4 + 1⁄x^4 + 2x² + 2⁄x².

Correct — A.
Group the expression: (x⁴ + 1⁄x⁴) + 2(x² + 1⁄x²)

Square the given sum: x² + 1⁄x² = (x + 1⁄x)² − 2 = 1 − 2 = −1
Square again: x⁴ + 1⁄x⁴ = (x² + 1⁄x²)² − 2 = 1 − 2 = −1

Substitute: −1 + 2 × (−1) = −3 → option (a)

Why the others are wrong

• (b)
  3 — 3 has the right size and the wrong sign. Both brackets come out as −1, so −1 + 2(−1) must be negative.
• (c)
  4 — 4 cannot come from this working: both brackets equal −1, so the expression is −1 − 2 = −3, a negative number.
• (d)
  −4 — −4 is one too low. Squaring subtracts 2 each time, giving x² + 1⁄x² = −1 and x⁴ + 1⁄x⁴ = −1, so the total is −3.

The reciprocal ladder: from x + 1⁄x = k, square to get x² + 1⁄x² = k² − 2, then square again to get x⁴ + 1⁄x⁴ = (k² − 2)² − 2.

The −2 is the cross term 2 × x × 1⁄x = 2. Group the target into these building blocks before substituting: here 2x² + 2⁄x² is simply 2(x² + 1⁄x²).

Key facts

• x² + 1⁄x² = (x + 1⁄x)² − 2
• x⁴ + 1⁄x⁴ = (x² + 1⁄x²)² − 2
• x² + 1⁄x² = (x − 1⁄x)² + 2
• If x + 1⁄x = −1, then x³ = 1

Common traps

• Adding 2 instead of subtracting it when squaring x + 1⁄x
• Rejecting a negative answer because x⁴ + 1⁄x⁴ looks as if it must be positive

No directly related past PYQ was found.