Show that the equation a/x-1 + b/x-2 + c/x-3 + d/x-4 =1 can't have imaginary roots if a,b,c,d are any four real numbers of the same sign.

we will use the method of contradiction
let us assume there be a complex root p+iq
after multiplying the conjugate in numerator and denominator we get
a{(p-1) - i q}/{(p-1)^2 + q^2)}  similarly we will write  all the three terms

there would be a term of q common in all the four terms as coefficient of i

now comparing l.h.s and r.h.s
 q( a/[(p-1)^2 + q^2)} + b/{(p-2)^2 + q^2)}+c/{(p-3)^2 + q^2)}+d/{(p-4)^2 + q^2)}]=0

imaginary part has to be 0 as a , b, c, d all are of same sign

the second  term will not be zero so q has to be zero
so our assumption was wrong there can''t be a imaginary root when a,b,c,d has
same sign