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Explanation:

Concept: Remainder of f(x) when divided by x - a is f(a).

Here f(x) = 4x¹⁰ - x⁹ + 3x⁸ – 5x⁷ + cx⁶ + 2x⁵ – x⁴ + x³ – 4x² + 6x – 2

It has to be divided by x - 1.

∴ Remainder will be f(1)

⇒ f(1) = 4 × 1¹⁰ - 1⁹ + 3 × 1⁸ – 5 × 1⁷ + c × 1⁶ + 2 × 1⁵ – 1⁴ + 1³ – 4 × 1² + 6 × 1 – 2 = 2
⇒ 4 - 1 + 3 – 5 + c + 2 – 1 + 1 – 4 + 6 – 2 = 2
⇒ c + 3 = 2
⇒ c = - 1

∴ c + 6 = -1 + 6 = 5

Hence, 5.