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

To determine the nature of the roots, we evaluate the discriminant using the formula b squared minus 4ac. For the given equation, the discriminant is the square of the quantity 3 plus b minus 4 times the quantity 2 plus b times the quantity 4 plus b. Expanding and simplifying this expression gives negative 3 times b squared plus 20 times b plus 23, which evaluates to a negative number for any real value of b. Since the discriminant is strictly less than zero, the quadratic equation has no real roots. The roots are imaginary.

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