Official solution
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.