If b∈F'then the roots of the equation (2+b)x2+(3+b)x+(4+b)=0 is

The given equation is

(2 + b) x^{2} + (3 + b) x + (4 + b) = 0

.
Comparing the values of the given equation with standard quadratic equation,

a x^{2} + bx + c = 0

we get the values of a, b, c as

a = 2 + b, b = 3 + b, c = 4 + b

.
Substituting the given values in the discriminant formula,

b^{2} - 4 ac

, we get,

(3 + b)^{2} - 4 (2 + b) (4 + b)

= b^{2} + 6 b + 9 - 4 b^{2} - 24 b - 32

= - 3 b^{2} - 18 b - 23

= - 3 (b^{2} + 6 b + 9) + 4

= - 3 (b + 3)^{2} + 9 < 0

So, roots are imaginary.
Hence, the correct option is 3.

If $b \in F^{'}$ then the roots of the equation $(2 + b) x^{2} + (3 + b) x + (4 + b) = 0 is$