**The correct option is: (C) -1, -6**

**Explanation:**

x^{4 }+ x^{3} + 8x^{2} + ax + b is exactly divisible by x^{2} + 1

=> Remainder must be zero.

(a - 1)x + (b - 7) = 0

=> a - 1 = 0 and b - 7 = 0 => a = 1 and b = 7

Now, ax^{2} + bx + 6 becomes x^{2} + 7x + 6.

x^{2} + 7x + 6 = x^{2 }+ 6x + x + 6 = 0

= x(x + 6)+1(x + 6) = 0

= (x + 1)(x + 6) = 0 => x = -1,-6