In trig form we have
#(R_1 e^(i theta_1))/(R_2 e^(i theta_2))#
#= (R_1)/(R_2) e^(i (theta_1- theta_2))#
#R_1 = sqrt ( (2)^2 + (-1)^2) = sqrt 5#
#R_2 = sqrt (4^2 + 1^2) = sqrt 17#
#tan theta_1 = - 1/2#
#tan theta_2 = 1/4#
From
#tan(alpha-beta) = (tanalpha-tanbeta)/(1+tanalphatanbeta)#
#tan (theta_1 - theta_2) =( (-1/2) - 1/4 ) / ( 1 + (-1/2)1/4 ) = - 6/7#
Which means that #cos (theta_1 - theta_2) = 7 / sqrt 85# and #sin (theta_1 - theta_2) = -6 / sqrt 85#
So
#(R_1 e^(i theta_1))/(R_2 e^(i theta_2))#
#= (sqrt 5)/(sqrt 17) e^(i (arctan -6/7))#
#= (sqrt 5)/(sqrt 17) ( 7 / sqrt 85 - i 6 / sqrt 85 ) #
#= 1/ (17) (7 - 6 i)#
It's a lot simpler by finding the complex conjugate of the denominator #d^prime# and multiply the whole thing by the by #(d^\prime) / (d^prime)#. As follows
#(2 - i)/(4 + i) * (4 - i)/(4 - i)#
#= (8 - 2i - 4 i -1)/(16 - 4i + 4 i + 1)#
#= (7 - 6 i)/(17)#
#= 1/ (17) (7 - 6 i)#
