Use missing output levels and a circle comparison to count roots

C has equation $y=(x^2+ax+4)/(x+b)$, with vertical asymptote x=−1 and a stationary point at x=2.

(i) Find a and b.

(ii) Find the other asymptote.

(iii) Without a calculator, find the set of values y cannot take.

(iv) Sketch C, showing axial intercepts, asymptotes and stationary points.

(v) Deduce the number of real roots of $(4-x^2)(x+1)^2=(x^2-4x+4)^2$.