x² + bx = 45 = 0 (b $$ \in $$ R)
has roots $$\alpha $$ + i$$\beta $$, $$\alpha $$ – i$$\beta $$
sum of roots = – b = 2$$\alpha $$
product of roots = 45 = $$\alpha $$² + $$\beta $$²

Let z = x + iy

$$ \therefore $$ |x + iy +1| = 2$$\sqrt {10} $$

$${\left| {x + iy + 1} \right|^2} = {\left( {2\sqrt {10} } \right)^2}$$

$$ \Rightarrow $$ (x + 1)² + y² = 40

$$ \Rightarrow $$ ($$\alpha $$ + 1)² + $$\beta $$² = 40

[putting real part $$\alpha $$ in place of x and imaginary part $$\beta $$ in place of y]

$$ \Rightarrow $$ $$\alpha $$² + 2$$\alpha $$ + 1 + $$\beta $$² = 40

$$ \Rightarrow $$ 45 + 2$$\alpha $$ + 1 = 40

$$ \Rightarrow $$ $$\alpha $$ = -3

$$ \therefore $$ -b = 2$$\alpha $$ = 2$$ \times $$(-3) = -6

$$ \Rightarrow $$ b = 6

By checking options we found b² – b = 30.