# How to determine difference between LHP poles and zeroes by observing Nyquist plot?

I am solving practice problems on Nyquist Plot and I got stuck at a problem having a solution which I do not get at all. Here it is:

Which of the following is the transfer function of a system having the Nyquist plot shown in figure below -

The options are as follows:
a) $$\\large \frac { K } { s ( s + 2 ) ^ { 2 } ( s + 5 ) }\$$

b) $$\\large \frac { K } { s ^ { 2 } ( s + 2 ) ( s + 5 ) }\$$

c) $$\\large \frac { K ( s + 1 ) } { s ^ { 2 } ( s + 2 ) ( s + 5 ) }\$$

d) $$\\large \frac { K ( s + 1 ) ( s + 3 ) } { s ^ { 2 } ( s + 2 ) ( s + 5 ) }\$$

Since at w = 0+, the phase is equal to -180 degrees, I concluded that it is a type 2 system i.e. 2 poles at origin, and thus option (a) was eliminated. I could not decide how to proceed further.

However, according to the solution given in the book, since the origin is encircled once, we have difference between number of poles and zeroes = $$\\frac { 360 ^ { \circ } } { 90 ^ { \circ } }\$$ = 4 and thus option (b) is correct.

I have following doubts -

1. Encirclement of origin can not tell difference between poles and zeroes if said poles and zeroes are not inside the nyquist contour, and here no option has any pole or zero in RHP.
2. Is it a method to divide total encircled angle with 90 degrees to find difference between LHP poles and zeroes?